Abstract. arXiv:2003.07342v1 [math.CO] 16 Mar 2020 · 2020. 3. 17. · bumpless pipe dreams and...

arX

iv:2

003.

0734

2v1

[m

ath.

CO

] 1

6 M

ar 2

020

BUMPLESS PIPE DREAMS AND ALTERNATING SIGN MATRICES

ANNA WEIGANDT

Abstract. In their work on the infinite flag variety, Lam, Lee, and Shimozono (2018)introduced objects called bumpless pipe dreams and used them to give a formula for dou-ble Schubert polynomials. We extend this formula to the setting of K-theory, giving anexpression for double Grothendieck polynomials as a sum over a larger class of bumplesspipe dreams. Our proof relies on techniques found in an unpublished manuscript of Las-coux (2002). Lascoux showed how to write double Grothendieck polynomials as a sum overalternating sign matrices. We explain how to view the Lam-Lee-Shimozono formula as adisguised special case of Lascoux’s alternating sign matrix formula.

Knutson, Miller, and Yong (2009) gave a tableau formula for vexillary Grothendieckpolynomials. We recover this formula by showing vexillary marked bumpless pipe dreamsand flagged set-valued tableaux are in weight preserving bijection. Finally, we give a bi-jection between Hecke bumpless pipe dreams and decreasing tableaux. The restriction ofthis bijection to Edelman-Greene bumpless pipe dreams solves a problem of Lam, Lee, andShimozono.

1. Introduction

Lascoux and Schutzenberger [LS82] introduced double Grothendieck polynomials, whichrepresent classes in the equivariant K-theory of the complete flag variety. The initial def-inition was in terms of divided difference operators, but in the intervening years, authorshave put forward many combinatorial models to study Grothendieck polynomials (see, e.g.,[FK94, KM05, LRS06]). One purpose of this article is to shine a spotlight on a lesser knownformula of Lascoux [Las02]. Lascoux’s formula realizes each Grothendieck polynomial as aweighted sum over alternating sign matrices (ASMs). Our goal is to make explicit connec-tions between Lascoux’s formula and subsequent work in the literature.

In the context of back stable Schubert calculus, Lam, Lee, and Shimozono [LLS18] intro-duced bumpless pipe dreams (BPDs) and used them to give a formula for double Schubertpolynomials. We extend this formula to the K-theoretic setting and give a bumpless pipedream formula for double Grothendieck polynomials. This formula is closely related to Las-coux’s formula in terms of ASMs. There is a natural, weight preserving bijection betweenbumpless pipe dreams and ASMs. Indeed, bumpless pipe dreams are transparently in bijec-tion with the osculating lattice paths of statistical mechanics. We connect Lascoux’s ASMformula to bumpless pipe dreams by observing that the key of an ASM is the same as theDemazure product of its corresponding bumpless pipe dream (see Theorem 4.3).

The proof of the K-theoretic bumpless pipe dream formula follows the outline set outin [Las02] in terms of ASMs. These techniques translate in a natural way to BPDs. We

Date: March 17, 2020.Key words and phrases. Grothendieck polynomials, bumpless pipe dreams, alternating sign matrices.

1

http://arxiv.org/abs/2003.07342v1

2 A. WEIGANDT

provide additional details which were omitted by Lascoux. The main ingredient is a tran-sition formula for double Grothendieck polynomials (see Theorem 2.3). Lascoux stated thisformula without proof. We provide one in Appendix A. The bumpless pipe dream formulafor Grothedieck polynomials follows by observing the weights on bumpless pipe dreams arecompatible with transition (see Proposition 5.2 and Lemma 5.3).

For the remainder of the introduction, we proceed with a summary of our main results.

1.1. Bumpless pipe dreams. Start with the six tiles pictured below.

(1.1)

These tiles are the building blocks for a network of pipes. We interpret each “plus” tile asa place where two pipes cross, one pipe running horizontally and the other vertically. Abumpless pipe dream is a tiling of the n ˆ n grid with the tiles in (1.1) so that

(1) there are n total pipes,(2) each pipe starts vertically at the bottom edge of the grid, and(3) pipes end horizontally at the right edge of the grid.

Write BPDpnq for the set of n ˆ n bumpless pipe dreams.Bumpless pipe dreams are in transparent bijection with certain osculating lattice paths

from statistical mechanics (see, e.g., [Beh08]). Indeed, osculating lattice paths have nearlythe same definition, though the crossing tile is often represented as an “osculating” tile .In the terminology of Lam-Lee-Shimozono, this tile is called a “bumping” tile. This is thereason for calling these new pipe dreams “bumpless.”

The bijection with osculating lattice paths yields a bijection between bumpless pipe dreamsand alternating sign matrices1. Here, we prefer the pipe dream interpretation of these objects,which considers each BPD to be a planar history of a (possibly non-reduced) word in thesymmetric group Sn. Thus, our use of the crossing tile is crucial.

Given P P BPDpnq, we write BpPq for the Demazure product of the column reading wordof P (see Section 2.3). Roughly, BpPq is the permutation obtained by tracing out the pathsof each pipe. Furthermore, if a pair of pipes crosses more than once, we ignore all crossingsafter the first. See Figure 1.

Fix w P Sn and let

Pipespwq “ tP P BPDpnq : BpPq “ wu.

A bumpless pipe dream is reduced if any pair of pipes crosses at most once. Write

RPipespwq “ tP P Pipespwq : P is reducedu.

Lam, Lee, and Shimozono showed that RPipespwq is connected by droop moves on BPDs.We define K-theoretic droops and show that Pipespwq is connected by droop moves combinedwith K-theoretic droops (see Proposition 4.5).

1This map was described in [BMH95, Figure 1]. The connection to osculating paths is explained in[Bra97]. See also [Beh08, Section 4].

BUMPLESS PIPE DREAMS AND ALTERNATING SIGN MATRICES 3

1 2 3 4 5 6 7

5

2

4

1

7

6

3

1 2 3 4 5 6 7

5

2

4

1

7

6

3

Figure 1. If P is the BPD pictured on the left, BpPq “ 5241763. The diagramon the right indicates which crossings should be ignored.

1.2. The bumpless pipe dream formula for β-double Grothendieck polynomials.

We work in the setting of the β-double Grothendieck polynomials of [FK94]. See Section 2.4for this definition. These polynomials represent classes in connective K-theory [Hud14]. Cer-tain specializations recover (double) Schubert and Grothendieck polynomials. In this section,we introduce the bumpless pipe dream formula for β-double Grothendieck polynomials.

Let

DpPq :“ tpi, jq : P has a tile in row i and column ju

and

UpPq :“ tpi, jq : P has a tile in row i and column ju.

Define xi ‘ yj “ xi ` yj ` βxiyj. We associate to P the weight

wtpPq “

¨˝ ź

pi,jqPDpPq

βpxi ‘ yjq

˛‚

¨˝ ź

pi,jqPUpPq

1 ` βpxi ‘ yjq

˛‚.

Theorem 1.1. The β-double Grothendieck polynomial for w P Sn is the weighted sum

Gpβqw px;yq “ β´ℓpwq

ÿ

PPPipespwq

wtpPq.

We prove Theorem 1.1 in Section 5.2.

Example 1.2. The elements of Pipesp2143q are pictured below.

4 A. WEIGANDT

Thus, applying Theorem 1.1, we see that

Gpβq2143px;yq “ px1 ‘ y1qpx3 ‘ y3q ` px1 ‘ y1qpx2 ‘ y1qp1 ` βpx3 ‘ y3qq

` px1 ‘ y1qpx1 ‘ y2qp1 ` βpx3 ‘ y3qq

` βpx1 ‘ y1qpx1 ‘ y2qpx2 ‘ y1qp1 ` βpx3 ‘ y3qq. ♦

As a corollary to Theorem 1.1, we obtain the following.

Theorem 1.3 ([LLS18]). The double Schubert polynomial for w P Sn is a sum over reducedbumpless pipe dreams:

Swpx;yq “ÿ

PPRPipespwq

ź

pi,jqPDpPq

pxi ´ yjq.

Example 1.4. Pictured below are the elements of RPipesp2143q.

Therefore, S2143px;yq “ px1 ´ y1qpx3 ´ y3q ` px1 ´ y1qpx2 ´ y1q ` px1 ´ y1qpx1 ´ y2q. ♦

We will also study marked bumpless pipe dreams, that is

MBPDpnq :“ tpP,Sq : P P BPDpnq and S Ď UpPqu.

Graphically, we represent pP,Sq by drawing P as usual and shading each cell which belongsto S. Write

MPipespwq “ tpP,Sq P MBPDpnq : P P Pipespwqu.

As an immediate corollary to Theorem 1.1, we have

Corollary 1.5. The β-double Grothendieck polynomial is the weighted sum

Gpβqw px;yq “

ÿ

pP,SqPMPipespwq

β |DpPq|`|S|´ℓpwq

¨˝ ź

pi,jqPDpPqYS

pxi ‘ yjq

˛‚.

For the reader familiar with pipe dreams (also known as RC-graphs) in the sense of [BB93,FK96, KM05], the BPD formulas for double Schubert and Grothendieck polynomials aregenuinely different from the pipe dream formulas. In small examples, such as w “ 132, wesee obstructions to finding a weight preserving bijection which explains the equality of theseexpressions (see Example 6.3). However, after specializing the y variables to 0, it should, inprinciple, be possible to find a weight preserving bijection from pipe dreams to marked BPDs.Finding a direct bijection is expected to be difficult. See Section 6 for further discussion.


1 1

23

4

Figure 2. Pictured on the left is a set-valued tableau in FSYTp14352q. Onthe right is its corresponding marked bumpless pipe dream.

1.3. Vexillary bumpless pipe dreams and flagged tableaux. A partition is a weaklydecreasing sequence of nonnegative integers λ “ pλ1, λ2, . . . , λkq. Write

Ypλq :“ tpi, jq : 1 ď j ď λi and 1 ď i ď ku.

A tableau of shape λ associates a positive integer to each pi, jq P Ypλq. We will dis-cuss semistandard tableaux, flagged tableaux, and set-valued tableaux in this section. SeeSection 2.6 for these definitions.

In Section 7, we restrict our attention to bumpless pipe dreams for vexillary permutations,i.e., those permutations which avoid the pattern 2143. We show Pipespvq “ RPipespvq if andonly if v is vexillary (see Lemma 7.2). Furthermore, if v is vexillary, Pipespvq is in transparentbijection with the sets of non-intersecting lattice paths studied by Kreiman [Kre05].

As a consequence of Kreiman’s work, we obtain a bijection

γ : FSYTpvq Ñ Pipespvq

where FSYTpvq is the set of flagged semistandard tableaux for v. Explicitly, the map takesa tableau T of shape µpvq to the unique P P Pipespvq so that

(1.2) DpPq “ tpT pi, jq, T pi, jq ` j ´ iq : pi, jq P Ypµpvqqu.

In particular, Pipespvq is in bijection with certain sets of excited Young diagrams [IN09], aswell as the (reduced) diagonal pipe dreams of [KMY09].

Continuing Kreiman’s lattice path story, we define a map

γ : FSYTpvq Ñ MPipespvq

where FSYTpvq is the set of flagged set-valued tableaux for v. If T P FSYTpvq, the minimumelement of each cell of T determines the diagram of the underlying BPD in the same wayas (1.2). The other elements indicate which upward elbow tiles are marked. We make thisprecise in Section 7.4. See Figure 2 for an example.

Theorem 1.6. Fix v P Sn so that v is vexillary. The map

γ : FSYTpvq Ñ MPipespvq

is a weight preserving bijection.

This bijection provides a new combinatorial proof of the formula for vexillary doubleGrothendieck polynomials given in [KMY09]. See Theorem 7.1.

6 A. WEIGANDT

6 5 4 2 1

4 3 2

3 2 1

1

Figure 3. The decreasing tableau on the left corresponds to the Hecke BPDP on the right. The tableau has reading word a “ p1, 3, 4, 6, 2, 3, 5, 1, 2, 4, 2, 1q.The reader may verify that Bpaq “ 5427136 “ BpPq.

1.4. Decreasing tableaux and Hecke bumpless pipe dreams. Write DTpn ´ 1q forthe set of decreasing tableaux with entries in t1, 2, . . . , n ´ 1u. Each decreasing tableau hasa column reading word formed by reading its labels within columns from bottom to top,starting at the left (see Figure 3).

Say that P P BPDpnq is a Hecke BPD if DpPq forms a northwest justified partition shape.Write HBPDpnq for the set of Hecke BPDs in BPDpnq. If DpPq corresponds to the partitionλ, say P has shape λ. If P is Hecke and reduced, then it is an Edelman-Greene BPD, asdefined in [LLS18].

We define a (shape preserving) map from HBPDpnq to DTpn ´ 1q as follows. First, no-tice that within any diagonal of the grid, P has as many crossing tiles as blank tiles (seeLemma 3.6). We pair the first blank tile within a diagonal with the last crossing tile, thenext with the second to last, and so on. With this convention, if a blank tile sits in row i

and its paired crossing tile in row i1, we assign the corresponding cell the value i1 ´ i. Write

Ω : HBPDpnq Ñ DTpn ´ 1q

for this map. See Figure 3 for an example.

Theorem 1.7. The map Ω : HBPDpnq Ñ DTpn ´ 1q is a shape preserving bijection. Fur-thermore, if P P Pipespwq, then the reading word of ΩpPq is a Hecke word for w.

The name Hecke BPD is meant to emphasize the relationship with the Hecke insertionof [BKS`08]. Each w P Sn has an associated formal power series Gwpx1, x2, . . .q, known asa stable Grothendieck polynomial. Buch [Buc02] proved we can write Gw as a finite linearcombination of stable Grothendieck polynomials corresponding to partitions:

Gw “ÿ

λ

awλGλ.

In particular, the coefficients are integers. Lascoux [Las01] showed these coefficients alternatein sign with degree. Buch, Kresch, Shimozono, Tamvakis, and Yong proved that |aw λ| countsincreasing tableaux of shape λ whose reading words are Hecke words for w. As a consequence


of Theorem 1.7 and [BKS`08, Theorem 1], |awλ| also counts the number of Hecke BPDs ofshape λ in Pipespwq.

Restricting the map Ω to Edelman-Greene BPDs provides a solution to [LLS18, Problem5.19]. This question was phrased in terms of increasing tableaux. However, we find BPDsto be naturally compatible with decreasing tableaux, so we focus on these objects instead.This is merely a convention shift (see [BKS`08, Section 3.8]). A previous bijection fromEdelman-Greene BPDs to (increasing) reduced word tableaux was given by Fan, Guo, andSun in [FGS18]. The statement of their bijection relies on Edelman-Greene insertion andthe Lascoux-Schutzenberger transition tree. Our bijection has the advantage that it is moredirect to state.

1.5. Organization. In Section 2, we recall the necessary background on reduced words inthe symmetric group and Grothendieck polynomials. In Section 3 we review relevant objectsfrom the literature which are in natural bijection with ASMs (and hence BPDs). We collectcertain facts for use in subsequent sections.

In the next two sections, we translate Lascoux’s work on ASMs into the language ofBPDs. Section 4 discusses keys of ASMs, Lascoux’s inflation moves, and K-theoretic droops.In particular, we show Lascoux’s notion of the key of an ASM is compatible with takingthe Demazure product of the associated BPD. In Section 5, we show that BPDs satisfy thetransition equations for Grothendieck polynomials. This allows us to prove Theorem 1.1.The foundation of the proofs in Section 5 was laid out by Lascoux, but we provide additionaldetails to make this discussion self-contained. We prove transition in Appendix A.

In Section 6, we recall (ordinary) pipe dreams (see [BB93, FK96, FK94, KM05]) andcompare the pipe dream formulas for Schubert and Grothendieck polynomials to the BPDformulas. Section 7 discusses vexillary Grothendieck polynomials. Finally, in Section 8, weshow Hecke BPDs are in shape preserving bijection with decreasing tableaux.

2. Background

2.1. The symmetric group. In this section, we review necessary definitions related tothe symmetric group. We refer the reader to [Man01] for further background. Write N “t0, 1, 2, . . .u and P “ t1, 2, . . .u. Define rns “ t1, 2, . . . , nu.

Let Sn be the symmetric group on n letters, i.e., the group of bijections from rns toitself. We often represent permutations in one-line notation, writing w “ w1w2 . . . wn wherewi :“ wpiq for all i P rns. We will also use cycle notation. We write w0 “ nn ´ 1 . . . 1 forthe longest permutation.

The Rothe diagram of w is the set

(2.1) Dpwq :“ tpi, jq : wi ą j, w´1j ą i for all i, j P rnsu.

The Coxeter length of w is ℓpwq :“ |Dpwq|. The Lehmer code of w is

cw “ pcwp1q, cwp2q, . . . , cwpnqq

where cwpiq “ |tj : pi, jq P Dpwqu|. The map w ÞÑ cw defines a bijection from Sn to

tpc1, c2, . . . , cnq P Nn : 0 ď ci ď n ´ i for all i P rnsu.

8 A. WEIGANDT

The Lehmer code is well behaved under the natural inclusion ι : Sn Ñ Sn`1; we havecιpwq “ pcwp1q, . . . , cwpnq, 0q. As such, we conflate these finite codes with sequences takingentries in N where all but finitely many terms are 0. This allows us to extend the mapw ÞÑ cw to S8.

A permutation v P Sn is vexillary if it avoids the pattern 2143, i.e., if there are no indices1 ď i1 ă i2 ă i3 ă i4 ď n so that vi2 ă vi1 ă vi4 ă vi3 . A permutation is dominant if itavoids the pattern 132, i.e., there are no positions 1 ď i1 ă i2 ă i3 ď n so that vi1 ă vi3 ă vi2 .A permutation is dominant if and only if its code is a weakly decreasing sequence.

2.2. Words in the symmetric group. In this section, we follow [KM04]. Write si for thetransposition pi i ` 1q and ti j for pi jq. The si’s satisfy braid relations

sisi`1si “ si`1sisi`1 for all i P rn ´ 2s

and commutation relations

sisj “ sjsi if |i ´ j| ą 1.

Furthermore, s2i “ id for each i P rn ´ 1s.A word for w is a tuple a “ pa1, . . . , akq such that w “ sa1 ¨ ¨ ¨ sak . This word is reduced

if k “ ℓpwq. The Demazure algebra is the free Z-module generated by tew : w P Snu withmultiplication defined by

(2.2) ewesi “

#ewsi if ℓpwsiq ą ℓpwq and

ew if ℓpwsiq ă ℓpwq.

Write ei :“ esi. The ei’s satisfy the same braid and commutation relations as the si’s:

eiei`1ei “ ei`1eiei`1 for all i P rn ´ 2s and eiej “ ejei if |i ´ j| ą 1.

Additionally, e2i “ ei.Given a word a “ pa1, . . . , akq with letters in rn ´ 1s, the Demazure product Bpaq is

the (unique) element of Sn so that ea1 ¨ ¨ ¨ eak “ eBpaq. If Bpaq “ w, we say that a is a Hecke

word for w. Using the relations among the ei’s, any word a can be simplified to a reducedword a1 such that Bpaq “ Bpa1q.

2.3. Planar histories. A northeast planar history P is a configuration of n pseudo-lines,constrained to a rectangular region which satisfy the following properties:

(1) each path starts at the bottom of the rectangle,(2) each path ends at the right edge of the rectangle,(3) paths move in a northward or eastward direction at all times, and(4) paths may cross at a point, but do not travel concurrently.

Bumpless pipe dreams are special cases of northeast planar histories.Suppose P has k crossings. We obtain a word from P as follows. Order the crossings

from left to right, breaking ties within columns by starting at the bottom and movingupwards. Label each crossing by counting the number of paths which pass weakly northeastof the crossing and then subtracting one. Write ai for the label of the ith crossing. ThenaP “ pa1, . . . , akq is the word of the planar history. We define wP “ sa1 ¨ ¨ ¨ sak . If aP is areduced word, we say P is reduced.


To compute wP graphically, label the paths at the bottom edge of the rectangle from leftto right with the numbers 1, 2, . . . , n. Extend these labels across crossings using the picturebelow.

(2.3)

j

ii

j

Then wPpiq is the final label of the path which ends in the ith row.Additionally, we may consider the Demazure product of aP . For brevity, we write BpPq :“

BpaPq. We may compute BpPq graphically in a similar way as before. However, at crossings,if i ă j, the labels change as pictured below.

(2.4)

j

ii

j

i

ij

j

Then BpPqpiq is the label of the path in the ith row at the right edge of the rectangle. Noticethat the second configuration in (2.4) occurs if and only if paths i and j have previouslycrossed, i.e., the planar history is not reduced. In particular, P is reduced if and only ifBpPq “ wP .

Lemma 2.1. Applying any of the local moves pictured below to P produces a new planarhistory P 1 so that wP “ wP 1 and BpPq “ BpP 1q.

(2.5) Ø Ø Ø

Furthermore, if P 1 was obtained from P by one of the moves below, then BpPq “ BpP 1q.

(2.6) Ø Ø

Proof. This is an immediate application of the labeling schemes from (2.3) and (2.4). Forthe replacements

Ø

there are six cases to check when computing BpPq. The verification is straightforward, so weleave this task to the reader.

The Rothe BPD for w is the (unique) BPD which has downward elbow tiles in positionspi, wpiqq for all i P rns and no upward elbow tiles. In other words, all of its pipes are hookswhich bend exactly once.

10 A. WEIGANDT

Lemma 2.2. If P is the Rothe BPD for w, then aP is a reduced word and BpPq “ wP “ w.

Proof. By assumption, all pipes of P are hooks. As such, pairwise, they cross at most once.Therefore, P is reduced (and so BpPq “ wP). Starting from the bottom edge of the grid,the pipe labeled i travels upwards and then turns right in cell pw´1piq, iq. Finally it hits theright edge of the grid in row w´1piq. Therefore, BpPqpw´1piqq “ i for all i P rns which impliesBpPq “ w.

2.4. β-double Grothendieck polynomials. We now recall the β-double Grothendieckpolynomials of [FK94]. Let R “ Zrβsry1, . . . , yns. We write Zrβsrx;ys :“ Rrx1, . . . , xns. Thesymmetric group Sn acts on Zrβsrx;ys by

w ¨ f “ fpxw1, xw2

, . . . , xwn;yq.

We define an operator πi which acts on Zrβsrx;ys by

πipfq “p1 ` βxi`1qf ´ p1 ` βxiqsi ¨ f

xi ´ xi`1

.

The operators πi satisfy the same braid and commutation relations as the simple reflectionsin Sn, as well as the following Leibniz rule:

(2.7) πipfgq “ πipfqg ` psi ¨ fqpπipgq ` βgq.

Notice that πipfq is symmetric in xi and xi`1. Furthermore, for any g, if f “ si ¨ f thenπipfgq “ fπipgq; setting g “ 1 yields πipfq “ ´βf . Thus, π2

i “ ´βπi.The β-double Grothendieck polynomials are defined as follows. For the longest per-

mutation, we set

Gpβqw0

px;yq “ź

1ăi`jďn

pxi ‘ yjq,

where xi ‘ yj “ xi ` yj ` βxiyj. If w P Sn with wi ą wi`1, define

(2.8) Gpβqwsi

px;yq “ πipGpβqw px;yqq.

Notice if wi ă wi`1 then Gpβqw px;yq is symmetric in xi and xi`1. Therefore,

(2.9) πipGpβqw px;yqq “ ´βGpβq

w px;yq.

Specializing β “ ´1, in Gpβqw px;yq yields the double Grothendieck polynomials. We

may further specialize each yi to 0 to obtain the (single) Grothendieck polynomial. Weget the double Schubert polynomials from β-double Grothendieck polynomials by setting

β to 0 and replacing each yi with ýi, i.e., Swpx;yq :“ Gp0qw px;ýq. Likewise, the (single)

Schubert polynomials are obtained by setting β and each of the yi’s equal to 0.

2.5. Transition equations. In this section, we recall Lascoux’s transition equations fordouble Grothendieck polynomials. Note that the statement of transition in [Las02] differsfrom our conventions here by a change of variables.

Say pi, jq is a pivot of pa, bq in w if:

(1) wpiq “ j,(2) i ă a and j ă b, and(3) if pi1, j1q P ri, as ˆ rj, bs ´ tpi, jq, pa, bqu then wpi1q ‰ j1.


The permutation w has a descent in position i if wi ą wi`1. Write

despwq “ maxpt0u Y ti : wi ą wi`1uq.

Notice despwq “ 0 if and only if w “ id.Fix a permutation w P Sn such that w ‰ id. Let a “ despwq and set

b “ maxtj : pa, jq P Dpwqu,

i.e., pa, bq is the rightmost cell in the last row of Dpwq. We call pa, bq the maximal corner

of w and denote it mcpwq. For the identity, the maximal corner is undefined. Let

φpwq “ ti : pi, jq is a pivot of mcpwq in wu.

If I “ ti1, . . . , iku with 1 ď i1 ă i2 ă ¨ ¨ ¨ ă ik ă a, write cpaqI for the cycle pa ik ik´1 . . . i1q.

Fix I Ď φpwq and write b1 “ w´1pbq. We define

wI “ wta b1cpaqI .

Notice ℓpwIq “ ℓpwq ` |I| ´ 1 (see Lemma 5.1).

Theorem 2.3. Keeping the above notation,

Gpβqw px;yq “ pxa ‘ ybqG

pβqwH

px;yq ` p1 ` βpxa ‘ ybqqÿ

IĎφpwq:I‰H

β |I|´1G

pβqwI

px;yq.

By making the appropriate specializations of Theorem 2.3, one immediately recovers tran-sition formulas for Schubert and Grothendieck polynomials. We state the version for doubleSchubert polynomials below.

Corollary 2.4. Given w P Sn with mcpwq “ pa, bq,

Swpx;yq “ pxa ´ ybqSwH`

ÿ

iPφpwq

Swtiupx;yq.

The statement of Theorem 2.3 is well known to experts (especially the case β “ ´1).For completeness, we include a proof in Appendix A. The proof for single Grothendieckpolynomials can be found in [Las01] and [Len03, Corollary 3.10]. Corollary 2.4 can beobtained from the double version of Monk’s rule in [KV97, Proposition 4.1]. See [KY04] fora graphical description of transition in terms of Rothe diagrams.

2.6. Partitions and tableaux. We will frequently refer to objects positioned in an n ˆ n

grid or the infinite grid P ˆ P. When represented graphically, the cell p1, 1q is positioned inthe northwest corner of the grid. We write pi, jq for the cell which is in the ith row from thetop and the jth column from the left.

Recall, a partition is a weakly decreasing tuple of nonnegative integers λ “ pλ1, λ2, . . . , λkq.We will often conflate λ with the subset of cells in the P ˆ P grid

Ypλq “ tpi, jq : i P rks and j P rλisu.

Write |λ| “ |Ypλq|.We now discuss tableaux. We refer the reader to [Ful97] for a reference. A tableau of

shape λ is a filling of Ypλq with elements of P, i.e., a function T : Ypλq Ñ P. We say T is

12 A. WEIGANDT

semistandard if its entries weakly increase along rows (reading left to right) and strictlyincrease along columns (reading top to bottom). Explicitly,

(2.10) T pi, jq ď T pi, j ` 1q and T pi, jq ă T pi ` 1, jq

whenever both sides of these inequalities are defined. Given a tuple f “ pf1, . . . , fkq, wesay T is flagged by f if T pi, jq ď fi for all pi, jq P Ypλq. Write FSYTpλ, fq for the set ofsemistandard tableaux of shape λ which are flagged by f .

A tableau is decreasing if its entries strictly decrease along rows and columns. WriteDTpnq for the set of decreasing tableaux which have entries in rns. Let DTpλ, nq be the setdecreasing tableaux of shape λ. Given T P DTpnq, we form its column reading word

by reading its entries along columns from bottom to top, working from left to right (seeFigure 3). Say T P DTpnq is a reduced word tableau if its column reading word isreduced. Write RWTpnq for set the reduced word tableaux in DTpnq. Likewise, we writeRWTpλ, nq for the reduced word tableaux of shape λ with entries in rns.

A set-valued tableau T is a filling of Ypλq with nonempty, finite subsets of P. Write

|T| “ÿ

pi,jqPYpλq

|Tpi, jq|

for the total number of entries in T.We may produce a tableau T from a set-valued tableau T by selecting a single entry

from each cell. In particular, define flattenpTq by flattenpTqpi, jq “ minpTpi, jqq for allpi, jq P Ypλq. Equivalently, this is the map from set-valued tableaux to ordinary tableauxwhich forgets all but the smallest entry in each cell.

We apply the adjective semistandard to T if each (ordinary) tableau which can obtainedby restricting entries in T is itself semistandard. In the same way, the definition of a flaggedtableau extends to the setting of set-valued tableaux. Let FSYTpλ, fq be the set-valuedsemistandard tableaux of shape λ which are flagged by f .

3. Bumpless pipe dreams through ice

As previously discussed, bumpless pipe dreams are in transparent bijection with oscu-lating lattice paths. These in turn, are in bijection with numerous objects from statisticalmechanics. See [Pro01] for a survey on alternating sign matrices and related objects. In thissection, we review the bijections between bumpless pipe dreams, alternating sign matrices,square ice configurations, and corner sum matrices. We also collect related facts.

3.1. Alternating sign matrices and Rothe diagrams. An alternating sign matrix

(ASM) is a square matrix with entries in t´1, 0, 1u so that within each row and column, thenonzero entries alternate in sign and sum to 1. Write ASMpnq for the set of ASMs of size n.

An ASM with no negative entries is called a permutation matrix. The subset of per-mutation matrices may be identified with Sn via the map Sn ãÑ ASMpnq which takes w P Sn

to the matrix which has 1’s in positions pi, wiq for each i P rns and zeros elsewhere.We often represent A P ASMpnq in an n ˆ n grid by plotting

(1) a filled dot ‚ in row i and column j if Ai j “ 1 and(2) an open dot ˝ in row i and column j if Ai j “ ´1.


¨˚˝

0 0 1 00 1 0 01 0 ´1 10 0 1 0

˛‹‹‚

Figure 4. Pictured to the left is A P ASMpnq. On the right is thegraph of A pictured with its defining line segments. The cells in DpAq “tp1, 1q, p1, 2q, p2, 1qu have been shaded gray for emphasis.

Call this the graph of A.We now describe the Rothe diagram of an ASM. Start from the graph of A P ASMpnq. For

each filled dot, send out a line segment to its right and below it. These segments terminatewhen they reach an open dot or the edge of the grid. Call these defining line segments

for A. The set of the coordinates of cells which do not contain any defining line segments isthe Rothe diagram2 of A, denoted DpAq. See Figure 4 for an example. Let

NpAq “ tpi, jq : Ai j “ ´1u.

The set DpAq corresponds to the positive inversions of A as defined in [RR86] while NpAqis the set of negative inversions.

We define a map Φ : ASMpnq Ñ BPDpnq by “smoothing” out bends in the definingsegments, i.e., by making the replacements pictured below.

ÞÑ ÞÑ

Lemma 3.1. The map Φ : ASMpnq Ñ BPDpnq is a bijection.

Proof. Bumpless pipe dreams are in transparent bijection with osculating lattice paths, whichare in turn in bijection with ASMs (see [Beh08, Section 4]). To change a BPD to an osculatingpath, simply replace each crossing tile with a bumping tile. Using this correspondence,Lemma 3.1 follows.

In light of this bijection, we say A P ASMpnq is reduced if ΦpAq is a reduced BPD.Furthermore, notice UpΦpAqq “ NpAq and DpΦpAqq “ DpAq. Thus, our weights on BDPsare compatible with the weights on ASMs found in [Las02], up to changing conventions.

3.2. Corner sums. The corner sum function of A P ASMpnq is

(3.1) rApi, jq :“iÿ

a“1

jÿ

b“1

Aa b.

By convention, we define rApi, jq “ 0 whenever i “ 0 or j “ 0. Write

Rpnq :“ trA : A P ASMpnqu.

2In [Wei17], the author studied Rothe diagrams of ASMs using a different convention. There, negativeinversions were included in the diagram. This alternative definition allows for a natural generalization ofFulton’s essential set to ASMs.

14 A. WEIGANDT

Lemma 3.2 ([RR86, Lemma 1]). Corner sums of n ˆ n ASMs are characterized by thefollowing properties.

(1) rApi, nq “ rApn, iq “ i for i P rns and(2) rApi, jq ´ rApi ´ 1, jq and rApi, jq ´ rApi, j ´ 1q are 0 or 1 for all i, j P rns.

The map A ÞÑ rA places ASMpnq and Rpnq in bijection. By [RR86], the explicit inversemap is obtained by setting

(3.2) Ai j “ rApi, jq ` rApi ´ 1, j ´ 1q ´ rApi ´ 1, jq ´ rApi, j ´ 1q.

Corner sum matrices have a natural poset structure defined by entry wise comparison.Say rA ď rB if rApi, jq ď rBpi, jq for all 1 ď i, j ď n. This induces a poset structure onASMs by declaring

A ď B if and only if rA ě rB.

The restriction of the poset ASMpnq to Sn produces the (strong) Bruhat order on thesymmetric group. Indeed, ASMpnq is the smallest lattice with this property [LS96].

We will also need notation for partial row and column sums of A. Write

(3.3) rowA,ipjq “jÿ

k“1

Ai k and colA,jpiq “iÿ

k“1

Ak j.

Notice that

(3.4) rApi, jq “iÿ

a“1

rowA,apjq “jÿ

b“1

colA,bpiq.

We call tpi, jq P DpAq : rApi, jq “ 0u the dominant part of DpAq (and DpΦpAqq). Noticeby Lemma 3.2, the dominant part of the diagram is always top-left justified and forms apartition shape.

3.3. Square ice. We now recall the bijection between ASMs and square ice configurations.An ice model is an orientation of the edges of the square lattice (or a subset thereof) so thatat each 4-valent vertex, two edges point inwards and two point outwards. At each 4-valentvertex, there are six possible configurations, pictured below.

(3.5)

As our underlying graph, take an pn` 2q ˆ pn` 2q square subset of the lattice. From this,remove the outer-most edges and the four corner vertices. Call this a framed square grid.The 1-valent vertices are boundary vertices and the edges which are adjacent to them areboundary edges. We refer to the other edges (and vertices) as being interior edges (andvertices). A square ice configuration (of size n) is an orientation of this graph so that

(1) horizontal boundary edges point inwards,(2) vertical boundary edges point outwards, and(3) at every interior vertex, two edges point in and two edges point out.


¨˚˝

0 0 1 00 1 0 01 0 ´1 10 0 1 0

˛‹‹‚

Figure 5. Pictured on the left is a square ice configuration I. The ASM onthe right is the image of I under the map defined by (3.6).

Write Icepnq for the set of square ice configurations of size n. To map from a square iceconfiguration to an ASM, replace each interior vertex with an element from t´1, 0, 1u asindicated below.

(3.6)

1 ´1 0 0 0 0

See Figure 5 for an example. This map defines a bijection from square ice configurations toASMs (see [EKLP92, Section 7]).

Just as there are six vertex states in square ice configurations, bumpless pipe dreamsconsist of six types of tiles. The bijection from square ice to ASMs, composed with thebijection from ASMs to BPDs, produces the following tile-by-tile correspondence.

(3.7)

Equivalently, the map draws pipes along edges which point to the left and the edges whichpoint down.

Lemma 3.3. The map defined by (3.7) is a bijection from square ice configurations tobumpless pipe dreams.

Remark 3.4. Defining segments of an ASM provide an easy description of the map ASMpnq ÑIcepnq. Orient horizontal (vertical) interior edges pointing to the left (downwards) if and onlyif they coincide with a defining line segment of the Rothe diagram. Due to (3.7) and thedescription of the map Φ, this is the inverse to the map given in (3.6).

The next lemma provides the explicit connection between square ice configurations andcorner sum functions.

Lemma 3.5. Take i, j P rns and write rApi, jq “ k. If A ÞÑ I, the restriction of rA toti ´ 1, iu ˆ tj ´ 1, ju corresponds to the interior vertex at pi, jq of I as pictured below.

16 A. WEIGANDT

”k ´ 1 k ´ 1

k ´ 1 k

ı ”k ´ 1 k

k k

ı ”k ´ 2 k ´ 1

k ´ 1 k

ı ”k k

k k

ı ”k ´ 1 k ´ 1

k k

ı ”k ´ 1 k

k ´ 1 k

ı

Proof. Suppose that A ÞÑ I. We have fixed rApi, jq “ k. Furthermore, since

rApi, jq ´ rApi ´ 1, jq P t0, 1u and rApi, jq ´ rApi ´ 1, jq P t0, 1u,

there are only six possibilities for„rApi ´ 1, j ´ 1q rApi ´ 1, jqrApi, j ´ 1q rApi, jq

.

These are precisely the types of matrices which appear in the statement of the lemma.By (3.2) and (3.6), we must have the correspondences pictured below.

ÐÑ

„k ´ 1 k ´ 1k ´ 1 k

ÐÑ

„k ´ 1 k

k k

From (3.6), we see that adjacent horizontal (and vertical) edges change direction at aninterior vertex pi, jq if and only if Ai j P t´1, 1u. Assume Ai j “ 0. Then

(1) rowA,ipjq “ 1 if and only if the adjacent horizontal edges to pi, jq in I point left, and(2) colA,jpiq “ 1 if and only if the adjacent vertical edges to pi, jq in I point down.

Since rApi, jq “ rApi ´ 1, jq ` rowA,ipjq, the horizontal edges point to the left if and only ifrApi´1, jq “ k´1. Similarly, the vertical edges point down if and only if rApi, j´1q “ k´1.These assertions are enough to determine the last four correspondences.

3.4. Additional properties of bumpless pipe dreams. We may now apply the abovebijections to derive facts about bumpless pipe dreams.

Lemma 3.6. A bumpless pipe dream is uniquely determined by knowing any of the following:

(1) the locations of its upward and downward elbow tiles,(2) the locations of its crossing and blank tiles, or(3) the locations of its horizontal and vertical tiles.

Proof. (1) This follows immediately from the bijection between ASMs and bumpless pipedreams.(2) Fix A P ASMpnq and let P “ ΦpAq. By Lemma 3.5,

(3.8) rApa, bq ´ rApa ´ 1, b ´ 1q “

$’&’%

0 if Pa b is blank

2 if Pa b is crossing, and

1 otherwise.

As such, rA can be uniquely recovered starting from the initial conditions rApi, 0q “ rAp0, iq “0 using (3.8). Since rA is determined by this information, so is P “ ΦpAq.


(3) By an analogous argument to the previous part, rA can be recovered by working alongantidiagonals, using the positions of the horizontal and vertical edges in ΦpAq. Explicitly,

(3.9) rApa ´ 1, bq ´ rApa, b ´ 1q “

$’&’%

´1 if Pa b is a horizontal pipe,

1 if Pa b is a vertical pipe, and

0 otherwise.

Thus, the result follows.

It is well known (say from the perspective of square ice) that there are as many crossingtiles as blank tiles in each BPD. Likewise, there are the same number of horizontal andvertical tiles (see [BDFZJ12, Section 2.1]). We will make use of the following refinement.

Lemma 3.7. Let P P BPDpnq.

(1) Within each diagonal of P, there are as many crossing tiles as blank tiles.(2) Within each antidiagonal of P, there are as many horizontal tiles as vertical tiles.

Proof. (1) Fix i ě 0. Then rAp0, iq “ 0 and rApn ´ i, nq “ n ´ i. Then by (3.8),

n ´ i “ rApn ´ i, nq ´ rAp0, iq “nÿ

k“i`1

rApk ´ i, kq ´ rApk ´ i ´ 1, k ´ 1q.

Since there are n ´ i summands, taking values in t0, 1, 2u, there are as many 0’s as 2’s.Therefore, P has as many blank tiles as crossing tiles in this diagonal. A similar argumentworks for the diagonal containing pi, 0q and pn, n ´ iq.(2) We have

0 “ rAp0, iq ´ rApi, 0q “i´1ÿ

k“0

rApi ´ 1 ´ k, 1 ` kq ´ rApi ´ k, 0 ` kq.

By (3.9), the summands take values in t´1, 0, 1u. Thus, there must be the same number of´1’s as 1’s in the sum. Likewise,

0 “ rApi, nq ´ rApn, iq “ní´1ÿ

k“0

rApn ´ 1 ´ k, i ` 1 ` kq ´ rApn ´ k, i ` kq.

Again, there are as many 1’s as ´1’s in the sum. Thus, in each antidiagonal, there are asmany horizontal tiles as vertical tiles.

Lemma 3.8. Fix A P ASMpnq and let P “ ΦpAq. The label of the crossing pi, jq P P isrApi, jq ´ 1.

Proof. Since pi, jq is a crossing tile,

rApi, jq “ rApi ´ 1, j ´ 1q ` 2.

Thus, the lemma follows from Lemma 3.5 by noting rApi ´ 1, j ´ 1q counts the number ofpipes which past strictly northwest of pi, jq.

18 A. WEIGANDT

ÞÑ

Figure 6. Pictured above is (a portion of) the graph an ASM before andafter applying inflation. The region from (4.1) has been outlined for emphasis.

4. Keys via bumpless pipe dreams

4.1. Inflation on alternating sign matrices. Lascoux [Las02] defined a procedure toiteratively remove the negative entries from an ASM. Fix A P ASMpnq and pick pa, bq Prns ˆ rns. The pivots of pa, bq are those positions pi, jq ‰ pa, bq so that

(1) pi, jq sits weakly northwest of pa, bq,(2) Ai j “ 1, and(3) the only nonzero entries of A in the region ri, rs ˆ rj, ss are Ai j and possibly Aa b.

Now suppose Aa b “ ´1. Write pi1, j1q, . . . , pik, jkq for the pivots of pa, bq. Furthermore,assume

i1 ă i2 ă ¨ ¨ ¨ ă ik and j1 ą j2 ą ¨ ¨ ¨ ą jk.

We say Aa b is a removable ´1 if there are no negative entries (excluding Aa b) in the region

(4.1)k´1ď

ℓ“1

riℓ, as ˆ rjℓ`1, bs.

In particular, if Aa b is removable, the only nonzero entries in this region happen at pa, bqand its pivots. A northeast most negative entry within A is always removable.

Inflation sets all of these entries to 0 and then places 1’s in the positions

pi1, j2q, . . . , pik´1, jkq.

See Figure 6 for an example. Applying inflation to a removable ´1 produces a well-definedASM with one less negative entry than the original ASM. If A ÞÑ A1 by inflation, also callthe map ΦpAq ÞÑ ΦpA1q inflation. The key of A, denoted κpAq, is the permutation obtainedby working along rows in A from left to right (starting with the top row) and removing eachremovable ´1 using inflation. See Figure 7 for an example.

Lascoux’s choice of the name key was meant to invoke an analogy with keys of semistan-dard tableaux. If we identify an ASM with its monotone triangle, the two definitions arecompatible [Ava10, Corollary 9]. Furthermore, an ASM is always (weakly) larger than itskey in the poset of ASMs, i.e., κpAq ď A [Ava10, Proposition 3].

Remark 4.1. Note that κpAq need not be maximal among permutations belowA. For instance

A “

¨˚˝

0 1 0 01 ´1 1 00 1 ´1 10 0 1 0

˛‹‹‚ą

¨˚˝

0 1 0 01 0 0 00 0 0 10 0 1 0

˛‹‹‚ą

¨˚˝

1 0 0 00 1 0 00 0 0 10 0 1 0

˛‹‹‚“ κpAq.


Figure 7. Above, we show the reduction of the graph of an ASM to thegraph of its key, 5241763. The ASM on the left corresponds to the BPDpictured in Figure 1.

In particular, this illustrates that if A ą B, it is not necessarily true that κpAq ą κpBq.

4.2. Keys of ASMs via Demazure products. In this section, we connect Lascoux’s keyto Demazure products of words defined by bumpless pipe dreams.

Lemma 4.2. Suppose A1 is obtained from A by removing a removable negative 1.

(1) BpΦpAqq “ BpΦpA1qq.(2) If the ´1 in A has k pivots, then |aΦpA1q| ` k ´ 2 “ |aΦpAq|.

Proof. (1) If the removable ´1 has just two pivots, inflation amounts to iteratively applyingthe moves in (2.5). (See the example below.)

Ñ Ñ Ñ Ñ

Otherwise, we can find a configuration as pictured below (with some number of irrelevanthorizontal / vertical black pipes).

Ñ Ñ Ñ

20 A. WEIGANDT

ÞÑ

Figure 8. Pictured above is deflation on an ASM, using the first and thirdpivots, (assuming the bottom right corner is in the diagram).

By Lemma 2.1, the above moves all preserve the Demazure product. At the level of ASMs,we are reduced to a negative entry with one less pivot. Continue in this way until thenegative entry has just two pivots. At this point, we are reduced to the previous argument.(2) From the proof of part (1), if k “ 2, the number of crossings is preserved. For eachadditional pivot beyond the second, a crossing is removed (due to the application of one ofthe moves pictured in (2.6)).

Theorem 4.3. Fix A P ASMpnq. Then BpΦpAqq “ κpAq.

Proof. By Lemma 2.2, BpΦpwqq “ w “ κpwq.If A P ASMpnq is not a permutation matrix, then there exists a sequence

A “ Ap0q ÞÑ Ap1q ÞÑ ¨ ¨ ¨ ÞÑ Ap|NpAq|q “ κpAq

where Apiq ÞÑ Api`1q is inflation of the leftmost ´1 in the topmost row of Apiq with negativeentries. Furthermore, by definition, κpAq “ κpApiqq for all i.

By repeated application of Lemma 4.2, BpΦpAqq “ BpΦpApiqqq for all i. Therefore,

BpΦpAqq “ BpΦpAp|NpAq|qqq “ κpAq.

Indeed, as long as a negative entry of A is removable, it does not matter what order weapply inflation.

Corollary 4.4. If A1 is obtained from A by applying inflation to a removable ´1 thenκpAq “ κpA1q.

Proof. By Lemma 4.2, BpΦpAqq “ BpΦpA1qq. Applying Theorem 4.3,

κpAq “ BpΦpAqq “ BpΦpA1qq “ κpA1q.

Corollary 4.4 implies [Las02, Corollary 4], which was stated without proof.

4.3. A procedure for generating BPDs. If A ÞÑ A1 is an inflation, we call the replace-ment A1 ÞÑ A deflation. Notice deflation corresponds to selecting pa, bq P DpAq and somesubset of the pivots of pa, bq. If A ÞÑ A1 by deflation, then call the map ΦpAq ÞÑ ΦpA1q defla-tion as well. If deflation uses exactly one pivot, we call it a simple deflation. Interpreted interms of bumpless pipe dreams, simple deflations are the droops of [LLS18]. See the example


below.

ÞÑ

It is permissible to have more (unpictured) pipes in the rectangular region, but none of thesemay include elbow tiles. By [LLS18, Proposition 5.3], any P P RPipespwq can be reachedfrom Φpwq by a sequence of droops.

We now introduce K-theoretic droops, which are moves of the form pictured below.

ÞÑ

ÞÑ

Again, there may be other unpictured pipes, but these must not have elbow tiles within thepictured rectangle.

Proposition 4.5. The set Pipespwq is closed under applying droops and K-theoretic droops.Furthermore, any P P Pipespwq can be reached from Φpwq by applying a sequence of droopsand K-theoretic droops.

Proof. That Pipespwq is closed under droops and K-theoretic droops follows from observingeach of these moves can be realized as a composition of the moves from Lemma 2.1.

We know by Theorem 4.3 and the definition of the key, starting from P P Pipespwq, thereexists a sequence of inflations taking P to Φpwq. Suppose P ÞÑ P 1 is an inflation. Supposefurther that there was a sequence of droops and K-theoretic droops from Φpwq to P 1.

Notice that each deflation move on a BPD can be realized as a composition of a droop,combined with some number of K-theoretic droops. Simply droop the first pivot (from thetop) involved in deflation, then apply K-theoretic droops to the rest in order.

We conclude with a lemma. Given P P BPDpnq, write

MutepPq “ď

pi,jqPDpPq

r1, is ˆ r1, js

for the mutable region of P. We write λpwq :“ MutepΦpwqq.

Lemma 4.6. Suppose P 1 is obtained from P by deflation. Then:

(1) MutepP 1q Ď MutepPq.

22 A. WEIGANDT

(2) Pi j “ P 1i j for all pi, jq R MutepPq.

(3) Given P P Pipespwq, we have Pi j “ Φpwqi j for all pi, jq R Ypλpwqq.

Proof. (1) By definition, it is immediate that deflation of pivots does not create any diagramboxes outside of MutepPq.(2) This is again clear, since deflation takes place entirely within MutepPq.(3) Since P P Pipespwq, we can reach P from Φpwq by a sequence of deflation moves. Thus,the claim follows by repeated application of (2).

4.4. Bumpless pipe dreams supported on a partition. Fix a partition λ. We writeBPDpλq for the set of tilings of Ypλq with the tiles pictured in (1.1) so that each pipe:

(1) starts in the bottommost cell of a column of Ypλq and(2) ends in the rightmost cell of a row of Ypλq.

Note that there is no requirement on the number of pipes which appear in the tiling. Wedefine sets DpPq and UpPq in the same way as before.

If P P BPDpnq and Ypλq Ď rns ˆ rns, restricting P to Ypλq produces a well-definedelement of BPDpλq. We write resλ for this map. Starting with P P BPDpλq, we would liketo complete it to a square BPD in a way which preserves DpPq and UpPq.

Lemma 4.7. Fix a partition λ Ď rms ˆ rns. Take P P BPDpλq and suppose there are k pipes

in P. There exists a unique rP P BPDpm`nq so that resλp rPq “ P and Dp rPq, Up rPq Ď Ypλq.

Proof. We start by constructing rP . Embed P in the pm ` nq ˆ pm ` nq grid so that it istop-left justified. Extend all of the horizontal and vertical pipes at the boundary edge ofYpλq so that they hit the right and bottom edges of the grid.

Let i be the index of the first row (from the top) which has a blank tile outside of Ypλq.Let j be the index of the left-most tile in row i so that pi, jq R Ypλq. In this cell, place adownward elbow tile. The only tiles to the right of position pi, jq are blank or vertical. Thus,we may extend the downward elbow to the right edge of the grid. Likewise, the only tilesbelow position pi, jq are blank or horizontal. So we extend the pipe to reach the bottom edgeof the grid.

Continue to the next row which has a blank tile outside of Ypλq and repeat the proceduregiven above. This construction results in a tiling which has no blank tiles or upward elbowsoutside of Ypλq. Furthermore, there are m` pipes, each connecting the bottom row of thegrid to the right column. Finally, a downward elbow was placed outside of Ypλq if and onlyif there was no pipe which exited λ in that row. Thus, a pipe exits every row. From this, wesee the construction produced a well-defined element of BPDpm ` nq.

Uniqueness follows from the bijection between ASMpm ` nq and BPDpm ` nq combinedwith the fact that ASMs are determined by the restriction of rA to the southeast most cornersof each connected component of NpAq Y DpAq (see [Wei17, Proposition 3.11]).

Often, this construction produces a BPD which larger than necessary. If the bottom rightcorner is a downward elbow, we are free to remove the last row and column to producea well-defined BPD of one size smaller. With this in mind, we define comppPq to be thesmallest possible square BPD which restricts to P. See Figure 9 for an example.


Figure 9. Let λ “ p3, 3, 2q. Pictured on the left an element P P BPDpλqand on the right is comppPq.

Lemma 4.8. Take P P BPDpλq. If comppP q P Pipespwq, then

Pipespwq Ď tcomppQq : Q P BPDpλqu.

Proof. Take P P BPDpλq. Let w “ BpcomppPqq. Notice, by construction, the tiles outsideof Ypλq in comppPq only depend on which pipes exit the southeast boundary of Ypλq in P.In particular, if P 1 P BPDpλq is obtained from P by applying inflation (or deflation), theircompletions agree outside of Ypλq.

By construction, UpcomppPqq Ď Ypλq. This means any inflation on comppPq happensentirely within Ypλq. As such, we apply a sequence of inflations to comppPq to reach Φpwq.Furthermore, each replacement along the way is confined to Ypλq. Thus, we see Ypλpwqq ĎYpλq and the result follows.

5. Transition on bumpless pipe dreams

This section follows the techniques outlined in [Las02], translated to the language ofbumpless pipe dreams. We provide additional details to make the arguments self-contained.Throughout this section, we adopt the notation from Section 2.5.

5.1. Preliminaries. Given P P Pipespwq, define mcpPq “ mcpwq. Notice mcpΦpidqq is notdefined. Define a map t : BPDpnq Ñ BPDpnq as follows. Let tpidq “ id. Otherwise, writepa, bq :“ mcpPq. We know since Pipespwq is connected by sequences of deflations, pa, bq is ablank tile or an upward elbow. If it is a blank tile, make the replacements pictured belowinvolving the blue pipes in the rectangle ra, w´1pbqs ˆ rb, wpaqs:

(5.1) ÞÑ .

If mcpPq is an upward elbow, make the following replacements:

(5.2) ÞÑ .

24 A. WEIGANDT

All other pipes in the region (not pictured in the diagrams) remain the same.

Lemma 5.1. Take w P Sn. Fix I Ď φpwq. Suppose P is obtained from Φpwq by deflatingthe pivots of mcpwq which sit in the rows of I. Then tpPq “ ΦpwIq. Furthermore, ℓpwIq “ℓpwq ` |I| ´ 1.

Proof. Write mcpwq “ pa, bq.If I “ H then no deflation occurred. As such, we are in the situation of p5.1q. At the

level of permutation matrices, we are simply swapping rows to remove the inversion mcpwqand so ΦpwHq “ tpPq.

Now suppose I ‰ H. Write I “ ti1, i2, . . . , iku with i1 ă i2 ă ¨ ¨ ¨ ă ik. Since P wasobtained from Φpwq by deflation of the pivots in I, we know P has downward elbows inpositions

(5.3) pi1, bq, pi2, wpi1qq, . . . , pik, wpik´1qq, pa, wpikqq.

Furthermore, P has a unique upward elbow tile which sits in mcpwq. This is removed whenapplying t. So tpPq is a Rothe BPD.

From the description in (5.3), we confirm tpPq “ ΦpwH ¨ cpaqI q “ ΦpwIq. That ℓpwIq “

ℓpwq ` |I| ´ 1 is a consequence of Lemma 4.2.

Below, we translate [Las02, Proposition 7] into the language of bumpless pipe dreams. Weinclude a proof for completeness.

Proposition 5.2. Fix a permutation w ‰ id. The restriction of t to Pipespwq defines abijection

rt : Pipespwq Ñď

IĎφpwq

PipespwIq.

Proof. Let pa, bq :“ mcpwq. Given I Ď φpwq, let PI be the BPD obtained from Φpwq byapplying deflation to the pivots of mcpwq in Φpwq which sit in the rows of I.Claim: rt is injective.

Fix P P Pipespwq. The only cell of P in Ypλpwqq affected by the map t is mcpwq. Thus,knowing the result of applying rt to P uniquely determines P.Claim: rt is well-defined.

Fix P P Pipespwq. We know there exists a sequence of deflations

P “ Pp0q ÞÑ ¨ ¨ ¨ ÞÑ Ppkq ÞÑ Φpwq.

We may assume, without loss of generality, that Ppkq “ PI for some I Ď φpwq. (In the caseI “ H, Ppkq “ Φpwq.) Furthermore, the sequence

tpPp0qq ÞÑ ¨ ¨ ¨ ÞÑ tpPpkqq

consists of the same deflation moves, since the transition replacements happen in a disjointregion of the grid from the deflations. By Lemma 5.1, tpPpkqq “ tpPIq “ ΦpwIq. Thus,tpPq P PipespwIq.Claim: rt is surjective.

Take P P PipespwIq. Then we know we can find a sequence of deflations so that

P “ Pp0q ÞÑ ¨ ¨ ¨ ÞÑ Ppkq ÞÑ ΦpwIq.


By Lemma 5.1, we know rtpPIq “ ΦpwIq. Furthermore, by Lemma 4.6, Ppiq agrees withΦpwIq on all tiles outside of YpλpwIqq. Then, we may make the same replacements in therectangle ra, w´1pbqs ˆ rb, wpaqs to get from Ppiq to some P 1piq as one does to go from ΦpwIqto PI . This produces a valid BPD.

We now claim P 1 P Pipespwq. To see this, notice

P 1 “ P 1p0q ÞÑ ¨ ¨ ¨ ÞÑ P 1pkq ÞÑ PI

is a valid sequence of deflations. Since PI P Pipespwq, so is P 1. Furthermore, it is immediatethat rtpP 1q “ P and so we are done.

Lemma 5.3. Fix w P Sn with mcpwq “ pa, bq and take P P Pipespwq. Then,

wtpPq “

#βpxa ‘ ybqwtptpPqq if tpPq P BPDpwHq, and

p1 ` βpxa ‘ ybqq wtptpPqq otherwise.

Proof. By the argument in the proof of Proposition 5.2, we see that tpPq P PipespwHq if andonly if mcpPq is a blank tile. So the result follows from (5.1) and (5.2) and the definition ofthe weights of BPDs.

5.2. Proof of Theorem 1.1. We now prove our main theorem.

Proof. The formula is trivial to verify for the case mcpwq “ p1, 1q. Fix w P Sn with mcpwq “pa, bq. Assume the bumpless pipe dream formula holds for all permutations v so that mcpvq ămcpwq in lexicographical order.

We have for all I Ď φpwq that mcpwIq ă mcpwq. By Theorem 2.3,

Gpβqw px;yq “ pxa ‘ ybqG

pβqwH


IĎφpwq:I‰H

β |I|´1G

pβqwI

px;yq.

Applying the inductive hypothesis,

Gpβqw px;yq “ pxa ‘ ybq

¨˝β´ℓpwHq

ÿ

PPPipespwHq

wtpPq

˛‚

` p1 ` βpxa ‘ ybqqÿ

IĎφpwq:I‰H

β |I|´1

¨˝β´ℓpwIq

ÿ

PPPipespwIq

wtpPq

˛‚

“ β´ℓpwqÿ

PPPipespwHq

βpxa ‘ ybqwtpPq

` β´ℓpwqÿ

IĎφpwq:I‰H

ÿ

PPPipespwIq

p1 ` βpxa ‘ ybqqwtpPq

“ β´ℓpwqÿ

PPPipespwq

wtpPq.

The last equality is a consequence of Proposition 5.2 and Lemma 5.3.

26 A. WEIGANDT

5.3. Specialization to double Schubert polynomials. We conclude by giving a newproof of the Lam-Lee-Shimozono formula for double Schubert polynomials.

Proof of Theorem 1.3. Fix P P Pipespwq. Then

β´ℓpwq wtpPq “ β´ℓpwq

¨˝

ź

pi,jqPDpPq

βpxi ‘ yjq

˛‚

¨˝

ź

pi,jqPUpPq

1 ` βpxi ‘ yjq

˛‚

“ β |DpPq|´ℓpwq

¨˝

ź

pi,jqPDpPq

pxi ‘ yjq

˛‚

¨˝

ź

pi,jqPUpPq

1 ` βpxi ‘ yjq

˛‚.

We know by Lemma 3.6 that |DpPq| “ |aP | ě ℓpwq. Therefore,

`β´ℓpwq wtpPq

˘|β“0 “

$&%

ź

pi,jqPDpPq

pxi ` yjq if |DpPq| “ ℓpwq, and

0 otherwise.

Since |DpPq| “ |aP |, we know |DpPq| “ ℓpwq if and only if aP is a reduced word, i.e., P is areduced bumpless pipe dream. Thus,

Swpx;yq “ Gp0qw px;ýq “

ÿ

PPRPipespwq

ź

pi,jqPDpPq

pxi ´ yjq.

6. Comparisons between ordinary pipe dreams and bumpless pipe dreams

We now discuss connections between BPDs and pipe dreams which appeared earlier inthe literature (see [BB93], [FK94], [KM05]). We call these objects ordinary pipe dreams todistinguish them from BPDs.

6.1. Ordinary pipe dreams. Write δpnq “ pn´1, n´2, . . . , 1q for the staircase partition.An ordinary pipe dream of size n is a tiling of Ypδpnqq with the tiles pictured below.

We form a planar history by placing upward elbow tiles in each of the cells

tpn, 1q, pn ´ 1, 2q, . . . , p1, nqu.

Any such tiling produces a network of n pipes. Each pipe starts at the left edge of the gridand ends at the top. We can define a reading word and Demazure product in an entirelyanalogous way as we did for bumpless pipe dreams. This time read crossings along rowsfrom right to left, starting at the top and working down. An ordinary pipe dream is reducedif pairwise pipes cross at most one time. We refer the reader to [KM04] for further details.

We identify each ordinary pipe dream with a subset S Ď Ypδpnqq by recording the positionsof the crossing tiles. Write BpSq for the Demazure product of the reading word of S. Define

wtpSq “ź

pi,jqPS

βpxi ‘ yjq.

With this weight, we recall the following theorem.


Theorem 6.1 ([FK94, KM04]). The β-double Grothendieck polynomial is the sum

Gpβqw px;yq “ β´ℓpwq

ÿ

S:BpSq“w

wtpSq.

Again, specializing β “ 0 and substituting y ÞÑ ý amounts to summing over reducedordinary pipe dreams with the usual weights. This gives a formula for double Schubertpolynomials.

6.2. Bijections from ordinary pipe dreams to bumpless pipe dreams. In light ofTheorem 1.1 and Theorem 6.1, we have two different sets of objects whose weight generatingfunctions produce the same polynomial. In particular:

Proposition 6.2. Marked BPDs for w are equinumerous with ordinary pipe dreams for w.

Proof. Consider Gp1qw p1; 0q. Under this specialization, each marked BPD and ordinary BPD

has a weight of 1. Applying Theorem 1.1 and Theorem 6.1, we see immediately that

Gp1qw p1; 0q “ |MPipespwq| “ |tS : BpSq “ wu|.

Indeed, one could hope for a weight preserving bijection which connects marked BPDsto ordinary pipe dreams. In the dominant case there is such a bijection, though for trivialreasons. When w is dominant,

(6.1) Gpβqw px;yq “

ź

pi,jqPDpwq

pxi ‘ yjq.

In particular, there is a single ordinary pipe dream and a single marked BPD, both of whichhave the same weight. Outside of the dominant case, there are clear obstructions to theexistence of a general weight preserving bijection, illustrated by the following example.

Example 6.3. The elements of MPipesp132q are pictured below.

We compare these to the ordinary pipe dreams for 132.

We have ordered each set of pipe dreams so that the x weights are the same from left to right.However, there is no bijection which preserves both x and y weights simultaneously. ♦

Adjusting expectations, once could hope for a direct bijection which respects the single x

weights (with the y variables specialized to 0). Even this seems difficult. Lenart, Robinson,and Sottile discussed the problem in [LRS06] saying:

28 A. WEIGANDT

There is one construction of double Grothendieck polynomials which we donot know how to relate to the classical construction in terms of RC-graphs.This is due to Lascoux and involves certain alternating sign matrices. Ourattempts to do so suggest that any bijective relation between these formulaswill be quite subtle.

Knutson [Knu19] attributes this difficulty to the fact that, while ordinary pipe dreams arenaturally compatible with the co-transition formula:

...one might expect it to be very difficult to connect the pipe dream formulato the transition formula, requiring “Little bumping algorithms” and the like,and essentially impossible if one wants to include the y variables.

Even if it is too much to hope for a bijective understanding of the double formulas, a directbijection from ordinary pipe dreams to marked bumpless pipe dreams which preserves the xweights would be highly interesting.

We note that there are bijections in the literature between (ordinary) reduced pipe dreamsfor vexillary permutations and flagged tableaux. A proof for permutations of the form 1kˆw0

is found in work of Serrano and Stump (see [SS12, Section 3.1]). More generally, Lenartstated a bijection (without proof) which holds for all vexillary permutations (see [Len04,Remark 4.12]). These, combined with Theorem 7.7, produce a bijection from reduced pipedreams for v to Pipespvq, when v is vexillary. Furthermore, this bijection preserves the x

weights.

7. Vexillary Grothendieck polynomials

In this section, we restrict our attention to vexillary permutations. There are many in-terconnected combinatorial formulas in the literature for expressing vexillary Schubert andGrothendieck polynomials. Notably, the flagged set-valued tableaux of Knutson, Miller, andYong [KMY09] provide a link between the work of Wachs [Wac85] on flagged tableaux andBuch [Buc02] on set-valued tableaux. Knutson, Miller, and Yong also gave geometric mean-ing to these tableaux by describing a weight preserving bijection with diagonal pipe dreams.Reduced diagonal pipe dreams naturally label irreducible components of diagonal Grobnerdegenerations of vexillary matrix Schubert varieties.

Diagonal pipe dreams are closely related to the excited Young diagrams of [IN09]. Inthe reduced case, they are complements of the families of non-intersecting lattice paths ofKreiman [Kre05]. We will see that these lattice paths are in transparent bijection withvexillary bumpless pipe dreams. We make this connection explicit in Section 7.3. We useKreiman’s lattice paths as a bridge between set-valued tableaux and vexillary bumpless pipedreams. Kreiman’s choice of flagging is different from the one we use here, but equivalent.We refer the reader to [MPP18, Section 3.3], in particular, Remark 3.10. See also [MPP17,Section 3].

7.1. Vexillary Grothendieck polynomials. If v is vexillary, write µpvq for the partitionobtained by sorting the entries of cv to form a weakly decreasing sequence. Equivalently,µpvq is the (unique) partition which has as many cells in each diagonal of Ypµpvqq as thereare elements of Dpvq in the same diagonal (see Lemma 7.3).


Recall, λpvq is the smallest partition so that Dpvq Ď Ypλpvqq. We define the flag f pvq by

setting fpvqi to be the maximum row index j so that pj, j ´ i`µ

pvqi q P Ypλpvqq. In other words,

this is row of the southeast most cell in Ypλpvqq which sits in the same diagonal as pi, µpvqi q.

Abbreviate FSYTpvq :“ FSYTpµpvq, f pvqq and FSYTpvq :“ FSYTpµpvq, f pvqq. We weight aset-valued tableau as follows:

wtpTq “ β |T|´|λ|ź

pi,jqPYpλq

ź

kPTpi,jq

xk ‘ yk`jí.

In Theorem 1.6, we will show if v is vexillary, there is a weight preserving bijection

γ : FSYTpvq Ñ MPipespvq.

We use this to recover a theorem of Knutson-Miller-Yong from the marked BPD formula forGrothendieck polynomials.

Theorem 7.1 ([KMY09]). If v P Sn is vexillary, then

Gpβqv px;yq “

ÿ

TPFSYTpvq

wtpTq.

Proof. Suppose v is vexillary. By Corollary 1.5,

Gpβqw px;yq “

ÿ

pP,SqPMPipespwq

β |DpPq|`|S|´ℓpwq

¨˝

ź

pi,jqPDpPqYS

pxi ‘ yjq

˛‚.

By Theorem 1.6, there is a weight preserving bijection

γ : FSYTpvq Ñ MPipespvq.

From this, the result follows immediately.

7.2. Vexillary bumpless pipe dreams. Write CpPq for the set of the coordinates of thecrossing tiles in P.

Lemma 7.2. Fix P P Pipespwq. Then P has a crossing tile in Ypλpwqq if and only if w isnot vexillary. Furthermore, if w is vexillary then RPipespwq “ Pipespwq.

Proof. We first prove the claim for Rothe bumpless pipe dreams. Suppose Φpwq has a crossingtile at pa, bq P Ypλpwqq. Then this crossing appears weakly northwest of some pc, dq P Dpwq.Furthermore, since Φpwq is a Rothe pipe dream, pa, bq must be strictly northwest of pc, dq. Tohave a crossing at pa, bq, necessarily w´1

b ă a and wa ă b. Furthermore, since pc, dq P Dpwq,c ă w´1

d and d ă wc. Thus, w´1b ă a ă c ă w´1

d and wa ă b ă d ă wc which implies w

contains 2143.Conversely, suppose w contains 2143. Then there exist 1 ď i1 ă i2 ă i3 ă i4 ď n so that

wi2 ă wi1 ă wi4 ă wi3 . In particular, this implies pi3, wi4q P Dpwq and pi2, wi1q P CpΦpwqq.Since i2 ă i3 and wi1 ă wi4 , it follows that pi2, wi1q P Ypλpwqq.

We now break the remainder of the argument into cases.Case 1: Suppose w is vexillary.

If P P Pipespwq is reduced, then |CpPq| “ |CpΦpwqq|. By Lemma 4.6, P agrees with Φpwqoutside of Ypλpwqq. Therefore, all crossings of P occur outside of the mutable region. In

30 A. WEIGANDT

particular, there are no K-theoretic droops available to apply to P. Thus by Proposition 4.5,we conclude RPipespwq “ Pipespwq.Case 2: Suppose w is not vexillary.

P agrees with Φpwq outside of Ypλpwqq. In particular, since Φpwq has less than ℓpwqcrossings outside of Ypλpwqq, so does P. Since P has at least as many crossing tiles as ℓpwq,it follows that P must have some crossing in Ypλpwqq.

Lemma 7.3. Each diagonal of Ypµpvqq has as many cells as the corresponding diagonal inDpvq.

Proof. If v is dominant, cv is already decreasing and the result is immediate.Whenever v is vexillary, but not dominant, there exists v1, also vexillary, so that Ypλpv1qq Ĺ

Ypλpvqq. We can obtain such a v1 by applying a droop move to a southeast most box of Φpvqand then “toggling transition,” i.e., performing the replacement pictured in (5.2).

Since v is vexillary, this droop does not involve crossing tiles. Thus it also preserves thecount of blank tiles within each diagonal. Thus, Dpv1q has the same number of cells in eachof its diagonals as Dpvq. Furthermore, cv and cv1 only differ by permuting two entries (theindices of the top and bottom rows of the droop). Thus, µpvq “ µpv1q.

Continuing in this way, we may construct a chain of vexillary permutations to reach adominant permutation.

For the discussion that follows, we will need a more local notion of a droop. Call thereplacements pictured below local moves.

(7.1) ÞÑ ÞÑ ÞÑ ÞÑ

Applying a local move to a BPD produces a new (well-defined) BPD. Furthermore localmoves do not modify crossing tiles. As such, if P ÞÑ P 1 by one of the moves in (7.1), itfollows that BpPq “ BpP 1q. We call the reversed replacements inverse local moves.

(7.2) ÞÑ ÞÑ ÞÑ ÞÑ

The following lemma says that for vexillary permutations, Pipespvq is connected by localmoves and inverse local moves. See Figure 10 for an example.

Lemma 7.4. Fix v P Sn so that v is vexillary.

(1) Any P P Pipespvq is uniquely determined by the positions of its blank tiles.(2) Pipespvq is closed under applications of the moves in (7.1). Furthermore, if P P

Pipespvq, there is a sequence

Φpvq “ P0 ÞÑ P1 ÞÑ ¨ ¨ ¨ ÞÑ Pk “ P

where each Pi ÞÑ Pi`1 is a local move.(3) There exists a unique T pvq P Pipespvq so that DpT pvqq “ Ypµpvqq. Call T pvq the top

bumpless pipe dream for v.(4) Any P P Pipespvq can be reached from T pvq by a sequence of inverse local moves.


Figure 10. The elements of Pipesp1432q are pictured above. The BPD onthe left is Φp1432q. The rightmost BPD is T p1432q. The arrows between BPDsindicate the application of a local move.

Proof. (1) By Lemma 3.6, each P P BPDpnq is uniquely determined by knowing the positionsof its crossing tiles and its blank tiles. By Lemma 7.2, for all P P Pipespvq, the crossing tilesoccur outside of Ypλpvqq. In particular, the crossing tiles are in the same location for eachP P Pipespvq. This implies P P Pipespvq is uniquely determined by the locations of its blanktiles.(2) By Lemma 7.2, since v is vexillary, there are no crossing tiles in Ypλpvqq. Furthermore,Pipespvq “ RPipespvq. As such, P can be reached from Φpvq by a sequence of droops.In particular, these droops do not involve crossing tiles. Therefore, each one is merely acomposition of the moves in (7.1).(3) If P is a Hecke BPD, then by Lemma 7.3, we are done. Otherwise, there exists somepi, jq P DpPq so that pi, jq is not in the dominant part of DpPq. By choosing a northwestmost tile in its connected component, we may assume WLOG that pi, jq has no blank tileimmediately to its left or immediately above. Since pi, jq is not in the dominant part ofDpPq, the tiles in pi ´ 1, jq and pi, j ´ 1q must be occupied by some pipe. In particular,pi ´ 1, j ´ 1q cannot be blank. Therefore, we are in one of the situations pictured in (7.1)and can apply a local move. We may continue this process until we have reached a HeckeBPD.

By the second part, Pipespvq is connected by local moves. In particular, this implies thateach diagonal contains the same number of blank tiles for each P P Pipespvq. As such, thereis at most one Hecke BPD in Pipespvq whose diagonal lengths are uniquely determined bythe number of cells in each diagonal of Dpvq. By Lemma 7.3, this partition coincides withµpvq.(4) This follows immediately from the construction of T pvq in the previous part. We maysimply reverse the path from P to T pvq by using inverse local moves.

32 A. WEIGANDT

7.3. Families of non-intersecting lattice paths. We now turn our attention to the fam-ilies of lattice paths from [Kre05]. For this section, place a point in the center of each cell ofthe n ˆ n grid. From this, create a square lattice. We will consider northeast lattice pathson this grid.

A northeast lattice path L is supported on λ if it:

(1) starts at the lowest cell of a column in Ypλq,(2) moves north or east with each step, and(3) ends at the rightmost cell of a row in Ypλq.

Suppose F “ pL1, . . . , Lkq is a tuple of northeast lattice paths supported on λ. We say F

is non-intersecting if no two pairs of paths in F share a common vertex. It is immediatethat there is an injection from tuples of non-intersecting northeast lattice paths supportedon λ into BPDpλq. The image is precisely

NCBPDpλq :“ tP P BPDpλq : P has no crossing tilesu.

As such, we conflate F with its image in NCBPDpλq from this point on. Furthermore,Kreiman’s ladder moves (named for the analogous moves of [BB93] on ordinary pipe dreams)are immediately seen to be the moves in (7.2).

Lemma 7.5. For any µ Ď λ, there exists a unique P P NCBPDpλq so that DpPq “ Ypµq.Write P

top

λ pµq for this BPD.

Proof. This is [Kre05, Lemma 5.3].

Write Pathsλpµq for the set of P P NCBPDpλq which can be reached from Ptop

λ pµq by asequence of inverse local moves.

Proposition 7.6. (1) If P P NCBPDpλq, then comppPq is a vexillary BPD.(2) The completion map defines a bijection from Pathsλpµq to Pipespvq for some vexillary

permutation v.

Proof. (1) Fix P P NCBPDpλq. By assumption, P has no crossing tiles. Therefore, comppPqdoes not have crossing tiles in MutepBpcomppPqqq. Therefore, by Lemma 7.2, BpcomppPqq isvexillary.(2) Write v “ BpPtop

λ pµqq. It is immediate that for all P P Pathsλpµq, the completioncomppPq P Pipespvq. Furthermore, by construction, the completion map is injective. Surjec-tivity follows from Lemma 4.8.

7.4. Tableaux to pipes. Recall the map γ : FSYTpvq Ñ Pipespvq which takes T P FSYTpvqto (the unique) P P Pipespvq so that

DpPq “ tpT pi, jq, T pi, jq ´ i ` jq : pi, jq P Ypµpvqqu.

Theorem 7.7. If v P Sn is vexillary, the map

γ : FSYTpvq Ñ Pipespvq

is a bijection.

Proof. This is a consequence of [MPP18, Proposition 3.6] and Proposition 7.6.


2 2

3

1 2

3

1 2

2

1 1

3

1 1

2

Figure 11. We list the flagged tableaux for v “ 1432. In this case, µpvq “p2, 1q, λpvq “ p3, 3, 2q, and f pvq “ p2, 3q.

In Figure 11, we list the elements of FSYTp1432q. They are positioned in the same relativeorder as their counterparts in Figure 10. Notice that applying a local move to P P Pipesp1432qcorresponds to decreasing the value of a cell within γpPq by one.

Now recallST “ tpk, k ´ i ` jq : k P Tpi, jq and k ‰ minpTpi, jqqu.

Define γ : FSYTpvq Ñ MPipespvq by

γpTq “ pγpflattenpTqq,STq.

We seek to show γ is a bijection. We start by introducing an auxiliary set of tableaux.Call T P FSYTpλ, fq saturated if for all pi, jq P Ypλq, there is no k ą minpTpi, jqq for whichadding k to Tpi, jq produces an element of FSYTpλ, fq. Write FSYTSpλ, fq for the set ofsaturated tableaux in FSYTpλ, fq. We abbreviate FSYTSpvq :“ FSYTSpµpvq, f pvqq.

Lemma 7.8. The map flatten : FSYTSpλ, fq Ñ FSYTpλ, fq is a bijection.

Proof. Take T,T1 P FSYTSpλ, fq so that flattenpTq “ flattenpT1q. Suppose for contra-diction, there exists k P Tpi, jqzT1pi, jq. Then k ď minpTpi, j ` 1qq “ minpT1pi, j ` 1qq,k ă minpTpi ` 1, jqq “ minpT1pi ` 1, jqq, and k ď fi. As such, we can add k to T1pi, jq toproduce a valid element of FSYTpλ, fq But then T1 is not saturated, which is a contradiction.An identical argument shows there is no k P T1pi, jqzTpi, jq. Thus, T “ T1.

We now show the map is surjective. Start with T P FSYTpλ, fq. Let T be the set-valued tableau so that Tpi, jq “ tT pi, jqu for all pi, jq P Ypλq. If T is saturated, the result isautomatic. If not, by definition, there is some pi, jq P Ypλq and k P P so that k ą minpTpi, jqqand adding k to Tpi, jq produces an element of FSYTpλ, fq. Continue this process, until theresulting tableau is saturated. This tableau maps to T by construction.

As an immediate consequence, FSYTSpvq is in bijection with Pipespvq. We now show theexcess entries in a saturated tableau for v tell us the positions of the upward elbows in thecorresponding BPD.

34 A. WEIGANDT

Lemma 7.9. If T P FSYTSpvq then ST “ UpγpflattenpTqqq.

Proof. First notice that

γ ˝ flatten : FSYTSpvq Ñ Pipespvq

is a bijection by Theorem 7.7 and Lemma 7.8.We proceed by induction on local moves from Φpvq. In the base case, let T P FSYTpvq

so that γpT q “ Φpvq. None of the entries of T can be increased without violating theflagging or semistandardness conditions defining FSYTpvq. In particular, this means T mapsto the saturated tableau T defined by Tpi, jq “ tT pi, jqu for all pi, jq P Ypµpvqq. Thus,ST “ H “ UpΦpvqq.

Now fix T P FSYTpvq. Let P “ γpT q. If P “ Φpvq we are done, so assume not. Then byLemma 7.4, there is P 1 P Pipespvq so that P 1 Ñ P is a local move. Let T 1 P FSYTpvq so thatγpT 1q “ P 1 and let T1 P FSYTSpvq so that flattenpT1q “ T .

Suppose ST1 “ UpP 1q. We seek to show ST “ UpPq. We proceed by analyzing the localmoves in (7.1). The effect of each local move T 1 ÞÑ T is to take some label k ` 1 “ T 1pi, jqand replace it with k. As such, Tpi, jq “ T1pi, jq Y tku. Furthermore, Tpi, j ´ 1q “ T1pi, j ´1q ´ tk ` 1u and Tpi ´ 1, jq “ T1pi, j ´ 1q ´ tku. All other entries of T and T1 agree.

These replacements are compatible with the local moves in (7.1). As such, ST “ UpPq.

We conclude by proving the main theorem of this section.

Proof of Theorem 1.6. That the map γ is weight preserving is immediate from the definitions.Furthermore, FSYTpvq and Pipespvq are in bijection (by Lemma 7.8 and Theorem 7.7.)

Elements of FSYTpvq are uniquely determined by first fixing T P FSYTSpvq and then freelyselecting a subset of the non-minimal elements of each cell. Likewise, elements of MPipespvqare determined by the choice of P P Pipespvq and a subset S Ď UpPq.

By Lemma 7.9, ST “ UpγpflattenpTqqq. Thus, the choice of non-minimal elements ineach cell uniquely determines a subset of UpPq (and vice versa).

8. Hecke bumpless pipe dreams

The goal of this section is to answer a problem posed in [LLS18] relating Edelman-GreeneBPDs to increasing tableaux whose row reading words are reduced words. We give a bijectionbetween Hecke BPDs and decreasing tableaux. In the case of reduced BPDs, this restrictsto a solution of [LLS18, Problem 5.19] (up to a convention shift).

8.1. Hecke bumpless pipe dreams. Recall, a Hecke BPD is a BPD whose diagram is topleft justified. In this case, the diagram necessarily forms a partition called the shape of theHecke BPD. Recall HBPDpnq is the set of Hecke BPDs of size n. Let HBPDpλ, nq be thesubset of Hecke BPDs of shape λ.

Lemma 8.1. Reflecting an ASM across its antidiagonal defines an involution on BPDs.Furthermore, this map has the effect of changing blank tiles to crossing tiles (and vice versa)and then reflecting these sets across the antidiagonal.


Proof. That the map is an involution is immediate. On square ice configurations, this maphas the effect of reversing arrows all arrows and then transposing across the antidiagonal.Translated to BPDs, this map changes crossings to blank tiles (and vice versa) and then flipsthese sets across the antidiagonal.

Given λ Ď δpnq, let vpλ, nq be the (unique) vexillary permutation so that

CpΦpvpλ, nqqq “ tpn ´ j ` 1, n ´ i ` 1q : pi, jq P Ypλqu.

Notice vpλ, nq is guaranteed to exist by Lemma 8.1, since there exists a dominant permutationu so that Dpuq “ Ypλq. Applying the above involution to u produces the Rothe BPD for avexillary permutation with the desired set of crossing tiles. We will make use of the followingrestriction of the involution on BPDs.

Lemma 8.2. There is a bijection between HBPDpλ, nq and Pipespvpλ, nqq.

Proof. When applying the map from Lemma 8.1, it is immediate that all Hecke tableaux ofshape λ map to BPDs for vpλ, nq and vice versa. Since this map is an involution on BPDpnq,the result follows.

Figure 12. Pictured above are the elements of HBPDpp2, 1q, 4q.

See Figure 12 for the Hecke BPDs which correspond to the elements of Pipesp1432q (pic-tured in Figure 11).

8.2. From flagged tableaux to decreasing tableaux. Pair blank tiles and crossing tileswithin diagonals by associating the first blank tile with the last crossing tile and so on. Nowgiven P P Pipespvpλ, nqq, define a tableau of shape λ by recording the difference of the rowindices of each crossing tile with its corresponding blank tile. Write

Γ : Pipespvpλ, nqq Ñ DTpλ, n ´ 1q

36 A. WEIGANDT

2 1

1

3 1

1

3 1

2

3 2

1

3 2

2

Figure 13. Above are the decreasing tableaux for vpp2, 1q, 4q “ 1432.

for this map. In Figure 13, we list the decreasing tableaux for vpp2, 1q, 4q “ 1432. They arein the same relative positions as their counterparts in Figure 11. Applying a local move toP P Pipesp1432q corresponds to increasing a label in ΓpPq by one.

Lemma 8.3. The map Γ : Pipespvpλ, nqq Ñ DTpλ, n ´ 1q is a bijection.

Proof. We start by showing Γ is well-defined. Take P P Pipespvpλ, nqq and write D “ ΓpPq.Fix T P FSYTpvpλ, nqq so that P “ γpT q.

Notice if P is the top BPD for vpλ, nq we have T pi, jq`Dpi, jq “ n´ j`1. In general, anylocal move on P has the effect of decreasing a label in T pi, jq and increasing the correspondinglabel in Dpi, jq (both by 1). Therefore, T pi, jq ` Dpi, jq “ n ´ j ` 1 for all pi, jq P Ypλq forarbitrary P P Pipespvpλ, nqq. In particular,

(8.1) Dpi, jq “ n ´ j ´ T pi, jq ` 1.

Since T is semistandard, it is quick to verify thatD is decreasing. Furthermore, by definition,Dpi, jq P rn ´ 1s (any diagonal in rns ˆ rns has at most n entries). Thus, D P DTpλ, n ´ 1q.

It is clear from (8.1), that the map is injective. All that remains is to show surjectivity.If P is the top BPD for vpλ, nq, then D is the maximal element in DTpλ, n ´ 1q, i.e.,Dpi, jq “ n ´ i ´ j ` 1 for all pi, jq P Ypλq. So it is enough to verify if an entry of Dcan decrease by 1 to produce a valid D1 P DTpλ, n ´ 1q, then there is a correspondingP 1 P Pipespvpλ, nqq so that P 1 ÞÑ D1.

Suppose Dpi, jq can be decreased. This means T pi, jq can be increased without violatingsemistandardness. Write pa, bq for the position of the blank tile in P which corresponds topi, jq P Ypλq. Since T pi, jq can be increased, pa` 1, bq, pa, b` 1q, and pa` 1, b` 1q are all notblank tiles in P (this follows from semistandardness). If pa ` 1, b ` 1q is a crossing tile, thiswould imply Dpi, jq “ 1. However this would mean Dpi, jq could not be decreased in thefirst place. So pa ` 1, b ` 1q is also not a crossing tile. Thus, we can apply an inverse localmove to P and we obtain the desired P 1 P Pipespvpλ, nqq.


8.3. Proof of Theorem 1.7. We now recall the map

Ω : HBPDpnq Ñ DTpn ´ 1q.

Fix P P HBPDpnq to DTpn ´ 1q. In each diagonal, pair the first blank tile with the lastcrossing tile, the next with the second to last, and so on. If a cell in DpPq sits in row i andits corresponding crossing tile in row i1, we fill the cell with the label i1 ´ i to obtain ΩpPq.

Notice that if P P HBPDpλ, nq, then ΩpPq is the result of composing the map fromHBPDpnq to Pipespvpλ, nqq and the map Γ : Pipespvpλ, nqq Ñ DTpλ, n ´ 1q. As such, Ω iswell-defined.

Lemma 8.4. The mapΩ : HBPDpnq Ñ DTpn ´ 1q

preserves commutation classes of reading words.

Proof. We will induct on n. In the base case n “ 1, there is nothing to show. Now fix n ą 1and suppose the statement holds for n ´ 1.

Take P P HBPDpnq and let T “ ΩpPq. Let A be the ASM which corresponds to P. LetT 1 be the tableau obtained by removing the first column of T . Let P P HBPDpn´ 1q so thatΩpP 1q “ T 1.

Write aP for the reading word of P. By the inductive hypothesis, the reading wordsof P 1 and T 1 are in the same commutation class. So it is enough to show we may applycommutation relations to move the reflections in aP which correspond to the first columnof T to the start of the word (keeping their relative order). Label the positions of thesecrossings pi1, j1q, pi2, j2q, . . . , pik, jkq.

Fix pi, jq P tpi1, j1q, pi2, j2q, . . . , pik, jkqu. By Lemma 3.8, this corresponds to the re-flection srApi,jq´1. We know pi, jq appears first in its respective column. Furthermore,any crossing which appears earlier in the reading order, but does not belong to the settpi1, j1q, pi2, j2q, . . . , pik, jkqu, must be strictly northwest of pi, jq. If pi1, j1q is strictly north-west of pi, jq, then rApi1, j1q ď rApi, jq´2 (this is Lemma 3.5 combined with the fact that rankfunctions of ASMs are weakly increasing). Thus, the reflection for pi, jq is free to commutepast the reflection for pi1, j1q (by Lemma 3.8).

Proof of Theorem 1.7. By Lemma 8.4, the map preserves commutation classes. Therefore,it also preserves Hecke products. That the map is a shape preserving bijection follows fromLemma 8.2 and Lemma 8.3.

From Theorem 1.7, we obtain the immediate corollary:

Corollary 8.5. The restriction of Ω to Edelman-Greene tableaux in BPDpnq defines a bijec-tion to RWTpn ´ 1q. This bijection is shape preserving and takes Edelman-Greene tableauxfor w to reduced word tableaux for w.

This provides a new solution to [LLS18, Problem 5.19].

Appendix A. Transition equations for β-double Grothendieck polynomials

In this appendix, we establish transition equations for β-double Grothendieck polynomials.First, we recall some notation. If k P P, let 1k be the tuple p1, . . . , 1, 0, 0, . . .q which starts

38 A. WEIGANDT

with k ones. If I “ ti1, . . . , iku, with 1 ď i1 ă i2 ă ¨ ¨ ¨ ă ik ă a, then cpaqI “ pa ik ik´1 . . . i1q.

Furthermore, if w P Sn and I Ď rns, write w ¨ I :“ twi : i P Iu.

Lemma A.1. Fix I Ď ra ´ 1s.

(1) If i R I or i ` 1 R I then sicpaqI si “ c

paqsiÏ

.

(2) If i, i ` 1 P I then sicpaqI “ c

paqI´ti`1u and c

paqI si “ c

paqI´tiu.

The proof of Lemma A.1 is elementary, so we omit it.Recall, φpwq “ ti : pi, jq is a pivot of mcpwq in wu. We typically write mcpwq “ pa, bq

and b1 “ w´1pbq. (In particular, a “ despwq.) Given I Ď φpwq, define wI “ wta b1cpaqI .

Lemma A.2. Fix w P Sn´1 and let a “ despwq. Suppose a ď k ă n and let w1 be (theunique) permutation in Sn so that cw1 “ 1k ` cw. Then

Gpβqw1 px;yq “

˜kź

i“1

xi ‘ y1

¸G

pβqw px1, . . . , xn´1; y2, . . . , ynq.

This lemma follows immediately from the ordinary pipe dream formula3.

Lemma A.3. Let w P Sn. Suppose wi “ 1 for some i P rdespwq ´ 2s. Let v “ wsi. Ifi R φpvq or i ` 1 R φpvq then

(1) φpwq “ si ¨ φpvq and

(2) for all I Ď φpvq, we have πipGpβqvI q “ G

pβqwsiÏ .

Proof. Throughout, write pa, bq “ mcpwq and let b1 “ w´1pbq.(1) Dpvq and Dpwq only differ in rows i and i ` 1. Since i ` 1 ă despwq, mcpvq “ mcpwq.Case 1: If i ` 1 R φpvq, then there is some j P φpvq with j ‰ i ` 1 so that pj, vjq Pri ` 1, as ˆ r1, bs. As such, pj, vjq “ pj, wjq P ri, as ˆ r1, bs and so i R φpwq.

i

i ` 1

j

a

wi wj b

i

i ` 1

j

a

vi`1 vj b

Furthermore, i P φpvq if and only if i ` 1 P φpwq. No other pivots of pa, bq are created ordeleted by swapping rows i and i ` 1. Therefore, φpwq “ si ¨ φpvq.

3Lemma A.2 can also be proved directly from first principles, i.e., the following arguments are not logicallydependent on pipe dreams. Regardless, we omit the proof.


Case 2: Now suppose i ` 1 P φpvq. By hypothesis, since i ` 1 P φpvq, we must have i R φpvq.This implies vi ą b.

i

i ` 1

a

wi wi`1b

i

i ` 1

a

vi`1 vib

Again, no other pivots are impacted by exchanging rows i and i ` 1 and so

φpwq “ pφpvq Y tiuq ´ ti ` 1u “ si ¨ φpvq.

(2) If i ` 1 R I, then for all I Ď φpvq, vIpi ` 1q “ 1 and so vI has a descent at i.On the other hand, suppose i`1 P φpvq but i R φpvq. Then we have vpiq ą b. Furthermore,

for any I Ď φpvq we have vIpi ` 1q ď b ă vIpiq. Hence, vI has a descent at i.

In both cases, by (2.8), πipGpβqvI q “ G

pβqvIsi. Applying Lemma A.1, we see

vIsi “ wsita b1cpaqI si “ wta b1sic

paqI si “ wta b1c

paqsiÏ

“ wsiÏ .

Thus, πipGpβqvI q “ G

pβqwsiÏ .

Lemma A.4. Let w P Sn. Suppose wi “ 1 for some i P rdespwq ´ 2s. Let v “ wsi. Ifi, i ` 1 P φpvq then φpwq “ φpvq ´ tiu. Furthermore, given I Ď φpvq,

πipGpβqvI

q “

$’&’%

GpβqvI´tiu if i, i ` 1 P I,

´βGpβqvI if i R I and i ` 1 P I, and

GpβqwsiÏ if i ` 1 R I.

Proof. Write pa, bq “ mcpwq and let b1 “ w´1pbq.Since i, i` 1 P φpvq, if we swap rows i and i` 1 to obtain w, we see that pi` 1, wpi` 1qq P

ri, as ˆ r1, bs and so i R φpwq.

i

i ` 1

iℓ

a

wi wi`1 b

i

i ` 1

iℓ

a

vi`1 vi b

This swap does not affect other pivots. Therefore, φpwq “ φpvq ´ tiu.Now fix I Ď φpvq.

40 A. WEIGANDT

Case 1: Suppose i, i ` 1 P I.

In this case, vIpiq ą vIpi ` 1q and so πipGpβqvI q “ G

pβqvIsi. By Lemma A.1, since i, i ` 1 P I,

vIsi “ vta b1cpaqI si “ vta b1c

paqI´tiu “ vI´tiu.

Thus, πipGpβqvI q “ G

pβqvI´tiu .

Case 2: Assume i R I and i ` 1 P I.

We have vIpiq ă vIpi ` 1q. Thus by (2.9), πipGpβqvI q “ ´βG

pβqvI .

Case 3: Suppose i ` 1 R I.

If i ` 1 R I, then vIpi ` 1q “ 1. Thus vI has a descent at i. Therefore, πipGpβqvI q “ G

pβqvIsi.

By Lemma A.1,

vIsi “ wsita b1cpaqI si “ wta b1sic

paqI si “ wta b1c

paqsiÏ

“ wsiÏ .

Therefore, πipGpβqvI q “ G

pβqwsiÏ .

Lemma A.5. Suppose w P Sn so that mcpwq “ pa, bq and wa´1 “ 1. Let v “ wsa´1. Thenthe following statements hold.

(1) φpwq “ φpvq \ ta ´ 1u.

(2) If I Ď φpvq then vI “ wIYta´1u “ wIsa´1 and πa´1pGpβqvI q “ G

pβqwI

.

Proof. (1) First, notice that mcpvq “ pa ´ 1, bq.

i1

i2

i3

a ´ 1

a

wa´1 wi1wi2wi3 b

i1

i2

i3

a ´ 1

a

va vi1vi2vi3 b

Since wa´1 “ 1 and wa ą b, automatically a ´ 1 P φpwq. Because mcpvq “ pa ´ 1, bq,a ´ 1 R φpvq. Furthermore, for any i ă a ´ 1, i P φapwq if and only if i P φpvq. Thus,φpwq “ φpvq \ ta ´ 1u.(2) Take I Ď φpvq. Notice b1 :“ w´1pbq “ v´1pbq. Then we have

vI “ wsa´1ta´1 b1cpa´1qI “ wta b1sa´1c

pa´1qI “ wta b1c

paqIYta´1u.

Thus, vI “ wIYta´1u.

Also, sa´1cpa´1qI “ c

paqI sa´1 and so vI “ wta b1c

paqI sa´1 “ w

paqI sa´1. We know vIpaq “

pwIsa´1qpaq “ wIpa ´ 1q “ 1. Therefore, vI has a descent at a ´ 1 and so

πa´1pGpβqvI

q “ GpβqvIsa´1

“ GpβqwI

.

With these preliminaries, we now prove transition.


Proof of Theorem 2.3. We will first induct on n. There is nothing to check for S1. Now fixn ą 1 and assume transition holds for permutations in Sn´1.

Take w P Sn and let pa, bq “ mcpwq. As a secondary inductive hypothesis, assume transi-tion holds for permutations u P Sn such that ℓpuq ą ℓpwq. By direct computation, one mayverify transition holds for the base case of w0 P Sn.

If w is the identity, then there is nothing to show. So assume not. We proceed by caseanalysis.Case 1: Assume wi ą 1 for all i P ra ´ 1s.

We have cwpiq ě 1 for all i P ra ´ 1s. Then there exists v P Sn´1 so that cw “ cv ` 1a´1.By induction on n, transition holds for v. Write rx “ px1, . . . , xn´1q and ry “ py2, . . . , ynq.Subcase 1: cwpaq “ 1.

In this case, b “ 1 and pa, bq is in the dominant part of Dpwq. Therefore, φpwq “ H.Notice that cwH

“ cv ` 1a´1. Then applying Lemma A.2, we see

Gpβqw px;yq “

ãź

i“1

xi ‘ y1

¸Gvprx; ryq

“ pxa ‘ y1q

ã´1ź

i“1

xi ‘ y1

¸G

pβqv prx; ryq

“ pxa ‘ y1qGpβqwH

px;yq.

Subcase 2: cwpaq ą 1.Observe mcpvq “ pa, b ´ 1q. In particular, we have φpwq “ φpvq. Furthermore, for all

I Ď φpwq, notice cwI“ cvI ` 1a. By induction on n, transition holds for v. Applying

Lemma A.2, we see

Gpβqw px;yq “

ãź

i“1

xi ‘ y1

¸G

pβqv prx; ryq

“

ãź

i“1

xi ‘ y1

¸pxa ‘ ybqG

pβqvH

prx; ryq

`

ãź

i“1

xi ‘ y1

¸p1 ` βpxa ‘ ybqq

ÿ

IĎφpvq:I‰H

β |I|´1G

pβqvI

prx; ryq

“ pxa ‘ ybqGpβqwH


IĎφpwq:I‰H

β |I|´1G

pβqwI

px;yq.

Case 2: Assume there some i P ra ´ 2s such that wi “ 1.Set v “ wsi. Since ℓpvq “ ℓpwq ` 1, by induction, transition holds for v, i.e.,

Gpβqv “ pxa ‘ ybqG

pβqvH


IĎφpvq:I‰H

β |I|´1G

pβqvI

.

42 A. WEIGANDT

Since xa ‘ yb is symmetric in xi and xi ` 1, by applying πi to both sides, we obtain:

(A.1) Gpβqw “ pxa ‘ ybqG

pβqwH


IĎφpvq:I‰H

β |I|´1πipGpβqvI

q.

Subcase 1: Suppose i R φpvq or i ` 1 R φpvq. As an immediate application of Lemma A.3,

Gpβqw “ pxa ‘ ybqG

pβqwH


IĎφpvq:I‰H

β |I|´1G

pβqwsiÏ

.

Also by Lemma A.3, φpwq “ si ¨ φpvq. In particular,

tI : I Ď φpwqu “ tsi ¨ I : I Ď φpvqu

and so the result follows.Subcase 2: Assume i, i`1 P φpvq. Take I Ď φpvq so that i R I and i`1 P I. By Lemma A.3,

(A.2) β |I|´1πipGpβqvI

q ` β |IYtiu|´1πipGpβqvIYtiu

q “ β |I|´1p´βqGpβqvI

` β |IYtiu|´1G

pβqvI

“ 0.

Now, suppose I Ď φpvq so that i, i ` 1 R I. Then

(A.3) β |I|´1πipGpβqvI

q ` β |IYtiu|´1πipGpβqvIYtiu

q “ β |I|´1G

pβqwI

` β |IYti`1u|´1G

pβqwIYti`1u

.

Since, φpwq “ φpvq ´ tiu, applying (A.2) and (A.3) to (A.1) produces the desired equation.Case 3: Assume wa´1 “ 1.

Let v “ wsa´1. Since mcpwq “ pa, bq, we know mcpvq “ pa ´ 1, bq. Furthermore, φpwq “φpvq Y ta ´ 1u.

Since ℓpvq ą ℓpwq, by the inductive hypothesis, transition holds for v. Therefore,

Gpβqv “ pxa´1 ‘ ybqG

pβqvH

` p1 ` βpxa´1 ‘ ybqqÿ

IĎφpvq:I‰H

β |I|´1G

pβqvI

.

We have πa´1pxa´1 ‘ ybq “ 1 and so,

πa´1ppxa´1 ‘ ybqGpβqvH

q “ GpβqvH

` pxa ‘ ybqπipGpβqvH

q ` βpxa ‘ ybqGpβqvH

(by (2.7))

“ p1 ` βpxa ‘ ybqqGpβqwta´1u

` pxa ‘ ybqGpβqwH

(by Lemma A.5).

Furthermore, πa´1p1 ` βpxa´1 ‘ ybqq “ 0. By Lemma A.5,

πa´1pp1 ` βpxa´1 ‘ ybqqGpβqvI

px;yqq “ p1 ` βpxa ‘ ybqqpπa´1pGpβqvI

q ` βGpβqvI

q (by (2.7))

“ p1 ` βpxa ‘ ybqqpGpβqwI

` βGpβqwIYta´1u

q.

Gpβqw “ pxa ‘ ybqG

pβqwH

` p1 ` βpxa ‘ ybqqGpβqwta´1u

` p1 ` βxa ‘ ybqÿ

IĎφpvq:I‰H

β |I|´1pGpβqwI

` βGpβqwIYta´1u

q

“ pxa ‘ ybqGpβqwH

` p1 ` βxa ‘ ybqÿ

IĎφpwq:I‰H

β |I|´1G

pβqwI

(by Lemma A.5).

Therefore, transition holds for w.


Acknowledgments

This project was inspired by a talk of Thomas Lam which took place at the Ohio StateSchubert Calculus Conference (2018). The author is grateful to Zachary Hamaker and OliverPechenik for numerous discussions about bumpless pipe dreams. The author also benefitedfrom helpful conversations and correspondence with Sara Billey, William Fulton, Allen Knut-son, Thomas Lam, Karola Meszaros, Mark Shimozono, David Speyer, and Alexander Yong.

References

[Ava10] Jean-Christophe Aval, Keys and alternating sign matrices, Sem. Lothar. Combin. 59 (2007/10),Art. B59f, 13.

[BB93] Nantel Bergeron and Sara Billey, RC-graphs and Schubert polynomials, Experiment. Math. 2(1993), no. 4, 257–269.

[BDFZJ12] Roger E. Behrend, Philippe Di Francesco, and Paul Zinn-Justin, On the weighted enumeration of

alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 119 (2012),no. 2, 331–363.

[Beh08] Roger E. Behrend, Osculating paths and oscillating tableaux, Electron. J. Combin. 15 (2008),no. 1, 60.

[BKS`08] Anders Skovsted Buch, Andrew Kresch, Mark Shimozono, Harry Tamvakis, and Alexander Yong,Stable Grothendieck polynomials and K-theoretic factor sequences, Math. Ann. 340 (2008), no. 2,359–382. MR 2368984

[BMH95] Mireille Bousquet-Melou and Laurent Habsieger, Sur les matrices a signes alternants, vol. 139,1995, Formal power series and algebraic combinatorics (Montreal, PQ, 1992), pp. 57–72.

[Bra97] Richard Brak, Osculating lattice paths and alternating sign matrices, Proceeding of Formal PowerSeries and Algebraic Combinatorics, vol. 9, Citeseer, 1997, p. 120.

[Buc02] Anders Skovsted Buch, A Littlewood-Richardson rule for the K-theory of Grassmannians, ActaMath. 189 (2002), no. 1, 37–78.

[EKLP92] Noam Elkies, Greg Kuperberg, Michael Larsen, and James Propp, Alternating-sign matrices and

domino tilings. II, J. Algebraic Combin. 1 (1992), no. 3, 219–234.[FGS18] Neil JY Fan, Peter L Guo, and Sophie CC Sun, Bumpless pipedreams, reduced word tableaux and

Stanley Symmetric functions, preprint (2018), arXiv:1810.11916.[FK94] Sergey Fomin and Anatol N. Kirillov, Grothendieck polynomials and the Yang-Baxter equation,

Formal power series and algebraic combinatorics/Series formelles et combinatoire algebrique,DIMACS, Piscataway, NJ, 1994, pp. 183–189.

[FK96] , The Yang-Baxter equation, symmetric functions, and Schubert polynomials, Proceedingsof the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence, 1993),vol. 153, 1996, pp. 123–143. MR 1394950

[Ful97] William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cam-bridge University Press, Cambridge, 1997, With applications to representation theory and ge-ometry.

[Hud14] Thomas Hudson, A Thom-Porteous formula for connective K-theory using algebraic cobordism,J. K-Theory 14 (2014), no. 2, 343–369.

[IN09] Takeshi Ikeda and Hiroshi Naruse, Excited Young diagrams and equivariant Schubert calculus,Trans. Amer. Math. Soc. 361 (2009), no. 10, 5193–5221. MR 2515809

[KM04] Allen Knutson and Ezra Miller, Subword complexes in Coxeter groups, Adv. Math. 184 (2004),no. 1, 161–176.

[KM05] , Grobner geometry of Schubert polynomials, Ann. of Math. (2) 161 (2005), no. 3, 1245–1318.

[KMY09] Allen Knutson, Ezra Miller, and Alexander Yong, Grobner geometry of vertex decompositions

and of flagged tableaux, J. Reine Angew. Math. 630 (2009), 1–31.

44 A. WEIGANDT

[Knu19] Allen Knutson, Schubert polynomials, pipe dreams, equivariant classes, and a co-transition for-

mula, preprint (2019), 15 pages, arXiv:1909.13777.[Kre05] Victor Kreiman, Schubert classes in the equivariant K-theory and equivariant cohomology of the

Grassmannian, preprint (2005), 27 pages, arXiv:math/0512204.[KV97] Axel Kohnert and Sebastien Veigneau, Using Schubert basis to compute with multivariate poly-

nomials, Adv. in Appl. Math. 19 (1997), no. 1, 45–60.[KY04] Allen Knutson and Alexander Yong, A formula for K-theory truncation Schubert calculus, Int.

Math. Res. Not. (2004), no. 70, 3741–3756.[Las01] Alain Lascoux, Transition on Grothendieck polynomials, Physics and combinatorics, 2000

(Nagoya), World Sci. Publ., River Edge, NJ, 2001, pp. 164–179.[Las02] , Chern and Yang through ice, preprint (2002).[Len03] Cristian Lenart, A K-theory version of Monk’s formula and some related multiplication formulas,

J. Pure Appl. Algebra 179 (2003), no. 1-2, 137–158.[Len04] , A unified approach to combinatorial formulas for Schubert polynomials, J. Algebraic

Combin. 20 (2004), no. 3, 263–299.[LLS18] Thomas Lam, Seung Jin Lee, and Mark Shimozono, Back stable Schubert calculus, preprint

(2018), 63 pages, arXiv:1806.11233.[LRS06] Cristian Lenart, Shawn Robinson, and Frank Sottile, Grothendieck polynomials via permutation

patterns and chains in the Bruhat order, Amer. J. Math. 128 (2006), no. 4, 805–848.[LS82] Alain Lascoux and Marcel-Paul Schutzenberger, Structure de Hopf de l’anneau de cohomologie

et de l’anneau de Grothendieck d’une variete de drapeaux, C. R. Acad. Sci. Paris Ser. I Math.295 (1982), no. 11, 629–633.

[LS96] , Treillis et bases des groupes de Coxeter, Electron. J. Combin. 3 (1996), no. 2, Researchpaper 27, approx. 35.

[Man01] Laurent Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMSTexts and Monographs, vol. 6, American Mathematical Society, Providence, RI; SocieteMathematique de France, Paris, 2001, Translated from the 1998 French original by John R.Swallow, Cours Specialises [Specialized Courses], 3.

[MPP17] Alejandro H. Morales, Igor Pak, and Greta Panova, Hook formulas for skew shapes II. Combina-

torial proofs and enumerative applications, SIAM J. Discrete Math. 31 (2017), no. 3, 1953–1989.[MPP18] , Hook formulas for skew shapes I. q-analogues and bijections, J. Combin. Theory Ser. A

154 (2018), 350–405.[Pro01] James Propp, The many faces of alternating-sign matrices, Discrete models: combinatorics, com-

putation, and geometry (Paris, 2001), Discrete Math. Theor. Comput. Sci. Proc., AA, MaisonInform. Math. Discret. (MIMD), Paris, 2001, pp. 043–058.

[RR86] David P. Robbins and Howard Rumsey, Jr., Determinants and alternating sign matrices, Adv.in Math. 62 (1986), no. 2, 169–184.

[SS12] Luis Serrano and Christian Stump, Maximal fillings of moon polyominoes, simplicial complexes,

and Schubert polynomials, Electron. J. Combin. 19 (2012), no. 1, Paper 16, 18.[Wac85] Michelle L. Wachs, Flagged Schur functions, Schubert polynomials, and symmetrizing operators,

J. Combin. Theory Ser. A 40 (1985), no. 2, 276–289.[Wei17] Anna Weigandt, Prism tableaux for alternating sign matrix varieties, Preprint (2017), 33 pages,

arXiv:1708.07236.

(AW) Department of Mathematics, University of Michigan, Ann Arbor, MI 48109

E-mail address : [email protected]

Date post:	12-Oct-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

Abstract. arXiv:2003.07342v1 [math.CO] 16 Mar 2020 · 2020. 3. 17. · bumpless pipe dreams and...

Documents